A modified Johnson-Cook model for 10%Cr steel at elevated temperatures and a wide range of strain rates

Author’s Accepted Manuscript A modified Johnson-Cook model for 10%Cr steel at elevated temperatures and a wide range of strain rates Jianli He, Fei Chen, Bo Wang, Luo Bei Zhu www.elsevier.com/locate/msea

PII: DOI: Reference:

S0921-5093(17)31356-4 https://doi.org/10.1016/j.msea.2017.10.037 MSA35640

To appear in: Materials Science & Engineering A Received date: 23 November 2016 Revised date: 12 October 2017 Accepted date: 12 October 2017 Cite this article as: Jianli He, Fei Chen, Bo Wang and Luo Bei Zhu, A modified Johnson-Cook model for 10%Cr steel at elevated temperatures and a wide range of strain rates, Materials Science & Engineering A, https://doi.org/10.1016/j.msea.2017.10.037 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

A modified Johnson-Cook model for 10%Cr steel at elevated temperatures and a wide range of strain rates Jianli Hea,*, Fei Chenb, *, Bo Wanga, LuoBei Zhua (a. School of Materials Engineering, Shanghai University of Engineering Science, Shanghai 201620, China; b

. National Engineering Research Center for Die and Mold CAD, School of Materials Science and Engineering, Shanghai Jiao Tong University, Shanghai 200030, China)

Abstract A modified J-C model of 10%Cr steel, based on the original J-C model, is further developed to consider not only the coupling effects of strains, strain rates and temperatures, but also the mechanism of the hardening and the flow softening during whole deformation processes. The model can simulate multi-deformation processes including work hardening, recovery and recrystallization at the high temperature for use as the nuclear power equipments. To establish this new model, the compression tests at different temperatures (950 °C-1250 °C) and strain rates (0.001 s-1-0.01 s-1) were conducted and the observations of microstructure under various conditions were performed. The results indicate that the influences of the strain, strain rate and temperature on the strain hardening of material are not independent of each other, while many of them interact. Moreover, Compared with the original J-C model and experimental data, the modified J-C model has good predictability of the hot deformation behavior of material. Keywords: 10%Cr steel; Elevated temperature; Hot deformation behavior; JohnsonCook model; Flow stress *

Corresponding authors at: School of Materials Engineering, Shanghai University of Engineering Science, Shanghai 201620, China; National Engineering Research Center for Die and Mold CAD, School of Materials Science and Engineering, Shanghai Jiao Tong University, Shanghai 200030, China E-mail addresses: [email protected] (J. He), [email protected] (F. Chen)

1

1. Introduction In order to reduce the cost of the nuclear power plants, it is necessary to improve plant efficiency through increasing the steam pressure and temperature. This implies increasingly stringent requirements on the materials of construction. The 10%Cr steel, X12CrMoWVNbN.10.1.1(hereafter referred to as X12), is an advanced low alloy steel used as the ultra-supercritical steam turbine rotor and blade material of nuclear power generation. It can undergo high temperature deformation. But this steel contains a variety of alloy compositions leading to complicated phase transitions involving the transformation of ferrite, austenite and martensite, at the same time with different types of precipitation. Furthermore, the precipitation during hot deformation tended to cause the formation of micro-cracks and large particles, such as M23C6. There are lots of research on this type of steel[1-5]. Cui et al.[2] investigated microstructure evolution of the X12 during short-time(<400h) creep within the range of temperatures from 600 °C to 750 °C and the characteristics of precipitations such as M23C6 (M: Cr, Fe, Mo, W, etc.) type carbides, MX (M: V, Nb, etc. and X: C, N) type carbonitrides and hardening phases. Han et al.[4] focused on discussing the influence of precipitations on the material performance. Additionaly, the kinetic law of austenite grain growth in the X12 was studied by quantitatively measurement of the austenite grain size after isothermal austenitized treatment. However, the deformation behavior of the X12 at elevated temperatures is still unclear. And, the coupling effects of strain rates, temperatures and strains on the behavior of the material are seldom considered. It is crucial to research the behavior of hot deformation at high temperature for X12. This is because the designing structure of the equipments of nuclear power units and making the forming process mainly depend on the hot deformation behavior. Furthermore, relatively accurate results

2

of nonlinear structure analysis and process optimization of numerical simulation rely on the high accuracy of constitutive model[6, 7]. The consititutive model can quantitatively describe the whole deformation behavior of the material during hot forming, which reflects the softening and hardening mechanism: work hardening, recovery (dynamic and static) and recrystallization (dynamic and static). The deformation behavior of material at high temperatures is a complex process. It is not only affected by stress state, but also by the strains, strain rates and temperatures. At present, a lot of researches on establishing the constitutive model have been carried out[8-11]. The constitutive models are divided into three categories: empirical, semi-empirical and physically based. Physically based models consider the microscopic factors such as dislocation, dislocation slip thermodynamics and activation energy. This type

includes

the

Zerilli-Armstrong(ZA)[12]

model,

the

Preston-Tonks-

Wallace(PTW)[13], Cellular Automaton(CA)[14], etc.. But these models can not predict the softening part of the flow curves accurately[15]. Empirical and semi-empirical models are widely used to predict the flow behavior since it is prone to be established and having less the materials constants[16, 17]. For instance, the Johnson-Cook(J-C) model[18], Arrhenius model[19], Khan-Huang-Liang(KHL) model[20] and artificial neural network[21]. Compared to other models, the J-C model can predict the deformation behaviors of metal materials at high temperature such as the structural steel[22, 23], the aluminum alloys[24, 25] and Ti alloys[26]. Moreover, there are less material parameters and less computational time[27] with this model. Lin[28] ， Song[29] and Li et al.[30] established the modified J-C model based on the original J-C model for various materials by conducting the tensile tests or the compression tests at

3

different temperatures and strain rates, considering the couple effects of strain rates, strains and temperatures on the hot deformation of materials. And the validity of the modified J-C models was examined by tensile tests, compression tests and torsion tests. these results showed that the modified J-C model could accurately predict the hot deformation behavior. But they only consider the effects of work hardening and recovery. In other words, the models can not reflect the mechanism of the softening. Additionally, these modified J-C models are not applicable to materials for use as nuclear power equipments. In this study, firstly, the compressive tests of the X12 were conducted by using thermal physical simulator gleeble 3800 at different temperatures (950 °C, 1050 °C, 1150 °C, 1250 °C) and different strain rates (0.001 s-1, 0.01 s-1, 0.1 s-1). And then the observations of microstructure were perfomed. The couple effects of strain rates, strains and temperatures on the hot deformation behavior of material were investigated. Secondly, a modified J-C model of X12 was proposed considering the work hardening and softening during the whole deformation processes. Finally, the reliability of the modified J-C model was evaluated by using statistical methods comparing to the original J-C model.

2. Experimental procedure The forged X12 in this investigation was adopted for the hot compression tests. Its chemical composition was given in Table 1. The compression specimens were machined (see Fig. 1). The compression tests were conducted by using the Gleeble3800 thermo-mechanical simulator at various temperatures (950 °C- 1250 °C) and strain rates (0.001 s-1- 0.1 s-1). Before the compression tests, the samples were heated to

4

1250 °C with a 10 °C·s-1 heating rate and held for 3 mins to eliminate the temperature gradient inside the sample. Then the samples were cooled to the predetermined temperature at a rate of 10 °C·s-1 and held for 30 seconds to eliminate the deformation gradient. The maximum true strain was 0.8 by the end of the compression tests. In order to prevent oxidation of the specimen surface, Each sample was protected by argon. It was quenched in water immediately to retain the microstructures at elevated temperature after each compression test. Furtherly, the microstructural evolution of the X12 at different temperatures and strain rates was studied by observation under an optical microscopy (VHK-600K). Compressed samples were cut vertically, polished, and etched with solution (H2O 100ml, picric acid 2g, 5% solution of sodium dodecyl benzene sulfonate (SDBS) 50ml, 6 drops of hydrochloric acid). Table 1 Chemical compostition of the X12 (wt. %) C

Si

Mn

Cr

Mo

Ni

V

S

P

0.11~0.13 W

≤0.12 Al

0.4~0.5 N

10.2~10.8 Nb

1.0~1.1

0.7~0.8

0.15~0.25

≤0.005

≤0.012

0.95~1.05

≤0.01

0.045~0.06

0.04~0.06

Fig. 1 Dimensions of the compression specimen

3. Results

5

The true stress vs. the true strain curves of the X12 at different temperatures and strain rates were shown in Fig. 2. The shape of each stress-strain curve exhibits the characteristics of hardening and flow softening behavior. The flow stress increases sharply to stress saturation with increasing strain during the hardening. During the softening, it decreases to a steady-state stress after the flow stress is up to the peak stress. It could also be seen from Fig. 2 that the values of flow stresses increase as the strain rate increases at the same temperature. However, the values of flow stresses decrease as the temperature increases at the same strain rate. The whole deformation process consists of elastic, plastic and steady stages. During the elastic stage, the flow stress and the strain follow a linear correlation. But the flow stress is not proportional to the strain during plastic stage. During the steady stage, dynamic recovery (DRV) and dynamic recrystallization (DRX) are the main softening mechanisms. Fig. 3 showed the microstructures of as-received X12 and the X12 after deformation which were observed under an optical microscopy at the temperatures of 950 °C to 1250 °C and the strain rates of 0.001 s-1 to 0.1 s-1. The grains after deformation at various temperatures and strain rates were different from original. The undeformed grains were nonuniform. The biggest grain size was up to 85 µm, while the smallest grain size was up to 3 µm. However, the grain size in Fig. 3b was fine and uniform comparing to the undeformed grains, in which the average grain size was 48 µm. Fig. 3c showed that there were mixed fine grains of lathy and irregular polygon. It could also be noiced from Fig. 3b through Fig. 3d that the microstructure changed. Fine grains were observed after deformation at various temperatures and strain rates, indicating the occurrence of the DRX.

The sub-grains appeared at the grain boundaries as shown in Fig. 3d. 105

-1 0.1s

160

90

True Stress (MPa)

True Stress (MPa)

140 120

0.01s

100

-1

80 60

0.001s

40

-1

75

-1 0.01s

60 45

-1 0.001s

30 15

20 0 0.0

-1 0.1s

0.2

0.4

True strain

0.6

0 0.0

0.8

0.2

0.4

True strain

6

0.6

0.8

(b)

(a) 45 70

40

50 40

-1 0.01s

30

-1 0.001s

20

30

-1 0.01s

25 20 15

-1 0.001s

10

10 0 0.0

-1 0.1s

35

-1 0.1s

True Stress (MPa)

True Stress (MPa)

60

5 0.2

0.4 True strain

0.6

0 0.0

0.8

0.2

0.4 0.6 True strain

(c)

0.8

(d)

Fig. 2 True stress vs. true strain curves of X12 obtained from compression tests: (a) 950 °C; (b) 1050 °C; (c) 1150 °C; (d) 1250 °C

(a)

(b)

(c) (d) Fig. 3 Microstructures of as-received X12 and the X12 after deformation at various conditions: (a)as-received; (b)950 °C, 0.001 s-1; (c) 1150 °C, 0.1 s-1; (d) 1250 °C, 0.1 s-1

7

4 Disscusion Fig. 2 and Fig. 3 revealed the hot deformation mechanism of X12. When the strain rate is kept constant, the flow stress drops with the increasing temperature at the same strain (see Fig. 2). This is because: 1) the thermal activation energy of the material and the average kinetic energy of the atom increase with increasing temperature, resulting in sufficient diffusion between atoms. 2) When the temperature increases, the critical shear stress decreases due to that a lower resolved shear stress is required to initiate a slip. 3) The fractions of dynamic recovery and recrystallization increase with increasing temperature, resulting in decreasing flow stress[8]. When the temperature is kept constant, the flow stress increases with increasing strain rate at the same strain, in which the dislocation is prone to migrate resulting from increasing the rate of dislocation accumulation. Besides, strain hardening is dominant in the plastic deformation stage, where the dislocation propagation causes the stress to increase rapidly. With increasing strain, the pile-up of sufficient dislocations storage during the hardening stage caused dynamic softening. The DRV is mainly controlled by dislocation climbing and sliding. Higher dislocation density indicates that the alloy in the zone has undergone severe deformation[31]. When a dynamic equilibrium exists between dislocation multiplication and annihilation, the flow stress remains stable during the steady stage. The microstructural investigations (see Fig. 3) show that deformation at low temperature leads to the increase in dislocation density. At this stage, the mechanism of strain hardening is dominant. The strain-hardening trend is related to precipitation characteristics. It is because that the precipitates on the grain boundaries and within the grain hinder the dislocation slip. Then the performance of material was enhanced from a macroscopic perspective. Furthermore, the strain hardening occurred from a

8

microcosmic perspective. At higher temperature, the softening mechanisms are dominant. Moreover, the appearance of fine grains is attributed to increased nucleation density. Additionally, Fig. 3 demonstrates that: a) the recrystallized fraction was carried out sufficiently at the low strain rate; b) it gives good plasticity due to fine and uniform distribution of grains at the relative high strain rate and at the temperature of 1150 °C. In conclusion, the temperature, strain rate and strain are the main factors that influence the material behavior, which interact with each other during whole deformation processes. In other words, the strain hardening and softening are strongly dependent on the coupling effects of the temperature and the strain rate.

5 The flow stress model 5.1 The J-C model In the section 4, the mechanism of the hot deformation behavior was discussed. Based on it, the constitutive model of the X12 was established by J-C model in this section. The J-C model was used to predict the behavior of material during hot deformation, which considered the influences of temperatures, strain rates and strains on the behavior of the material [32, 33]. It was defined as

   A  B n 1  C ln   1  T m 

(1)

where σ is Von Misses flow stress, ε is the equivalent plastic strain,

    0 is the dimensionless strain rate with   being the strain rate and  0 the reference strain rate,

9

T* =  T-Tr / Tm -Tr  with T as the current absolute temperature, Tm the melting temperature, and Tm is equal to 1520 °C for the present material; Tr as the reference temperature. In present investigation, 950 °C is taken as the reference temperature and 0.1 s-1 the reference strain rate. The J-C model has five material constants which need to be determined by a few experiments. A is the yield stress at the reference temperature and the reference strain rate; B and n are the coefficient of strain hardening; C and m are the material constants which represent the coefficient of strain rate hardening and thermal softening exponent, respectively. 5.1.1 Determination of material constants A, B and n When the temperature and the strain rate were equal to Tr and  0 respectively, Eq. (1) was rewritten as

  A B n

(2)

According to Fig. 2, the value of A could be obtained from the yield stress of the flow curve, A=74.12Mpa. Taking natural logarithm of Eq. (2), Eq. (3) was derived as follows

ln   Aln Bn ln 

(3)

Then substituting the value of A, the values of true stress vs. strain at various strain rates and temperatures into Eq. (3), the relationship between ln(σ-A) and lnε could be given in Fig. 4. The values of B and n were obtained from the slope and intercept of the fitting line in the ln(σ-A) vs. lnε plot, namely, B=129.83, n=0.4812. 5 4

lnε

3 2 1 0

-6.0

-4.5

-3.0 10 ln(σ-A)

-1.5

0.0

Fig. 4 Relationship of ln(σ-A) with lnε

5.1.2 Determination of material constant C When the temperature kept a constant, Eq. (1) was described as

 A  B n

 1  C ln  

(4)

According to Eq. (4), the values of C were obtained from the slope of the fitting lines in σ/(A+Bεn)- ln   plot, in which sixteen values of C corresponding to the different strains (0.1-0.7) were evaluated by linear fitting method in Fig. 5. Then the average value of C was equal to 0.1059. 1.20 0.01 0.06 0.11 0.16 0.21 0.26 0.31 0.36

1.05

σ/(A+Bεn)

0.90 0.75

0.41 0.46 0.51 0.56 0.61

0.66 0.71 0.78

0.60 0.45 0.30 -5

-4

-3

ln  

-2

-1

0

Fig. 5 Relationships between σ/(A+Bεn) and ln   to determine the value of C

5.1.3 Determination of material constant m When the strain rate kept a constant, Eq. (1) was rewritten as

1



 A  B  n

 T m

(5)

Taking natural logarithm of both sides of Eq. (5), Eq. (6) was determined as follows

11

   ln 1    m ln T    A  B n    

(6)

Substituting the values of the corresponding flow stresses at different strains into Eq. (6), the relationships between ln(1-σ/(A+Bεn)) and lnT* were obtained in Fig. 6. Then the constant m was determined from the slope of the averaged fitting line, in which its value was equal to 0.579. -0.15

ln(1-σ/(A+Bεn))

-0.30 -0.45 -0.60 -0.75 0.01 0.06 0.11 0.16 0.21 0.26 0.31

-0.90 -1.05 -1.20 -1.35 -1.8

-1.6

-1.4

-1.2

lnT

0.36 0.41 0.46 0.51 0.56 0.61 0.66 -1.0

-0.8

0.71 0.78

-0.6

*

Fig. 6 Relationships between ln(1-σ/(A+Bεn)) and lnT* to determine the value of m

In conclusion, the original J-C model for the X12 was established as follows

   74.12  129.83 0.4812 1  0.1059ln   1  T 0.579 

(7)

Comparisons between the experimental and predicted data by Eq. (7) were shown in Fig. 7. This figure showed that the original J-C model could not correctly predicted the flow behavior of the X12 at elevated temperatures. Especially, the prediction was bad under the deformation conditions of higher temperatures. This is because the thermal softening, strain rate hardening and strain hardening are three independent phenomena and can be isolated from each other in the J-C model [34, 35]. However, as a matter of fact, the constant m and C, namely, the thermal softening parameter m will be affected by the coupling of the strain and strain rate. And the strain rate sensitivity parameter C will be affected by the coupled of the strain, temperature and strain rate, which can be

12

observed from Figs. 9 through 12. Besides, the strain hardening and softening are affected by the coupling of the temperature and strain rate, which will be represented by a coefficient related to temperature and strain rate. Therefore, it is necessary to modify the original J-C model for describing the high-temperature deformation mechanism of the materials for use as nuclear power equipments. Predicted

Experimented 105

160

0.1s

-1 90

-1 0.1s

75

120

0.01s

100

-1

80 60

0.001s

-1

40

True Stress (MPa)

True Stress (MPa)

140

-1 0.01s

60 45 30

-1 0.001s

15

20 0 0.0

0.2

0.4 True strain

0.6

0 0.0

0.8

0.2

0.4 True strain

0.6

0.8

(b)

(a) 45

70

40 35

-1 0.1s

50 40

True Stress (MPa)

True Stress (MPa)

60

-1 0.01s

30 20

-1 0.001s

10 0 0.0

-1 0.1s

30

-1 0.01s

25 20 15

-1 0.001s

10 5

0.2

0.4 0.6 True strain

0 0.0

0.8

0.2

0.4

0.6

0.8

True strain

(c)

(d)

Fig. 7 Comparisons between the experimental and predicted flow stresses by the original J-C model at the temperature of a) 950 °C; b) 1050 °C; c) 1150 °C; d) 1250 °C

5.2 Modified J-C model In this study, a modified J-C model was proposed considering the coupled effects of strains, temperatures and strain rates on the behavior of material. It was also taken

13

into account initial deformation stage, work hardening, dynamic recovery and dynamic recrystallization stage. The modified J-C model was presented as

  A1 n  1   b1  b2  b3 2  ln    exp  1  2  T   1

(8) where the parameter A1 is a coefficient that reflects the interaction of the strain, temperature and strain rate. b1, b2, b3, λ1 and λ2 are the material constants; the reference temperature Tr is 950 °C; the reference strain rate  0 is 0.1 s-1. In equation (8), the first term  n1 can reflect initial deformation stage. This is because the power function are accurately describe the flow behavior of steel at small strain [36]. Other terms in Eq. (8) can explain the strain hardening and softening. The different material constants in Eq. (8) were obtained by regression analysis based on the experimental values of the true stressstrains at various strain rates and temperatures (see Fig. 2). 5.2.1 Determination of material constants n1 When the temperature and the strain rate kept constants, Eq. (8) was described as

  A1 n

1

(9)

substituting the values of true stress-strain at the reference temperature of 950 °C and the reference strain rate of 0.1 s-1 into Eq. (9), the value of n1 was obtained in Fig. 8, Namely, n1=0.13.

14

160

True Stress (MPa)

140

120

Data points Power fit Relevance of 0.9648

100

80

60 0.0

0.2

0.4 True strain

0.6

0.8

Fig. 8 Relationship of true stress with true strain at at the reference temperature of 950 °C and the reference strain rate of 0.1 s-1

5.2.2 Determination of material constants A1, b1, b2 and b3 When the temperature kept a constant, Eq. (8) was expressed as

  1   b1  b2  b3 2  ln   n A1

(10)

1

Let D be equal to b1  b2  b3 2 , D  b1  b2  b3 2 , then Eq. (8) was described as

  A1  1  D ln    n

(11)

1

Relationships σ/(εn1) between ln   at different strains (strains from 0.1 to 0.7 with the interval of 0.05) were shown in Fig. 9. It could be seen from Fig. 9 that the slope and the intercept of the fitting line were the value of A1·D and A1, respectively. Then the values of D at various strains were obtained in Table 2. According to Table 2, the relationship of D with strain was acquired by least square method, as shown in Fig. 10. Fig. 10 showed that the values of b1, b2 and b3 could be evaluated as 0.0852, 0.1042, 0.0609, respectively. Besides, the values of A1 at various strain were obtained by appling the inverse algorithm and the method of averaging them, namely, A1=165.57.

15

0.01 0.06 0.11 0.16 0.21 0.26 0.31 0.36

180 160

σ/(εn1)

140 120

0.41 0.46 0.51 0.56 0.61 0.66 0.71 0.78

100 80 60 -5

-4

-3

ln  

-2

-1

0

Fig. 9 Relationships between σ/(εn1) and ln  

0.13

Value of D

0.12

0.11

0.10

0.09 0.00

0.15

0.30

0.45

0.60

0.75

Strain

Fig. 10 Correlation of D with ε at the reference temperature of 950 °C

Table 2 Values of the material constant D corresponding to the different strains ε

0.01

0.06

0.11

0.16

0.21

0.26

0.31

0.36

D

0.0888

0.0913

0.0965

0.099

0.1019

0.1062

0.1096

0.1144

ε

0.41

0.46

0.51

0.56

0.61

0.66

0.71

0.78

D

0.1184

0.1215

0.1241

0.1261

0.128

0.1286

0.1279

0.1266

5.2.3 Determination of material constants λ1 and λ2 When the strain rate kept a constant, Eq. (8) was expressed as



 A    1  b  b   b   ln   n1

1

2

1

2

3

16



 exp  1  2  T  

(12)

Let λ be equal to 1  2 ,   1  2 , and take the logarithm of both sides of Eq. (12), then Eq. (12) was rewritten as

  ln    A1 n1   1   b1  b2  b3 2  ln   





   T   

(13)

substituting A1=165.57, n1=0.13, b1=0.0852, b2=0.1042 and b3=-0.0609 into Eq. (13), the relationship between ln   A1 n   1   b1  b2  b3 2  ln     and T  was obtained   1

according to the values of the flow stress-strains at the reference temperature of 950 °C, as shown in Fig. 11. Then the values of λ at different strains were obtained from Fig. 11, in which λ relation with ε was acquired by linear fit in Fig. 12. The values of λ1 and λ2 were derived from Fig. 12, namely, λ1=-1.9365, λ2=-1.9648. In conclusion, the values of all parameters in modified J-C model were listed in Table 3.

Table 3 Values of different parameters Parameter Value

A1 165.57

n1

b1

b2

0.13

0.0852

0.1042

b3 -0.0609

λ1

λ2

-1.9365

-1.9648

0.2 0.0 -0.2 -0.4 0.01 0.06 0.11 0.16 0.21 0.26 0.31 0.36 0.41

-0.6 -0.8 -1.0 -1.2 -1.4 -1.6 -1.8 0.0

0.1

0.46 0.51 0.56 0.61 0.66 0.71 0.78 0.2

0.3

T

0.4

0.5

0.6

Fig. 11 Relationships between ln   A1 n   1   b1  b2  b3 2  ln     and T  at the reference temperature of 950 °C, the reference strain rate of 0.1 s-1 and different strains 1

17

-2.0

-2.2

λ

-2.4

-2.6

-2.8

-3.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

Strain

Fig. 12 Relationship of λ with ε at the reference temperature of 950 °C

5.2.4 Verification of the modified J-C model Considering A1=165.57, n1=0.13, b1=0.0852, b2=0.1042 and b3=-0.0609, λ1=-1.9365 and λ2=-1.9648, the modified J-C model of the X12 was expressed as:   165.57 0.13  1   0.0852  0.1042  0.0609 2  ln    exp  1.9365 1.9648 T  

(13)

Comparisons between the experimental and predicted data by the modified J-C model of the X12 were shown in Fig. 13. It could be seen in Fig. 13 that there was a reasonable agreement between the experimental and predicted results. However, as can be seen in Fig. 13d, the model prediction does not closely follow the experimental data at the temperature of 1250 °C and the strain rate of 0.001 s-1. The main reason for this deviation is that the hardening exponent in the model is invariable. But there exists a minus error. Thus the modified J-C model can reflect the behavior of strain hardening and softening. To further verify the predictability of modified J-C model, the modified J-C model and the original J-C model were compared using statistical methods– the normalized bias error(NMBE) at different temperatures and strain rates. The expression of NMBE is as follow:

18

RMSE 

1 N 2  Ei  Pi   i 1 N

(14)

where RMSE is the average root mean square error, Ei is the experimental data and Pi is the predicted value obtained from the model. N is the total number of data employed in the investigation. According to Eq.(14), RMSE of the modified and original J-C model at various temperatures and strain rates were obtained in Table 4. Table 4 showed that the RMSE of the modified J-C model was much smaller than that of original J-C model, especially at low temperature and strain rate. For example, the RMSE of the modified J-C model and original model were 6.93 Mpa and 18.32 Mpa at the temperature of 950 °C and the strain rate of 0.001 s-1 , respectively. This implied that the modified J-C model had better the predictability of the flow behavior of the X12 during deformation processes. This is because that the modified J-C model considered not only the coupling effects of strains, strain rates and temperatures, but also strain hardening and strain softening. Predicted

Experimented

160

100

-1 80

100

0.01s

80

-1

60

0.001s

-1

True Stress (MPa)

0.1s

120

0.1s

60

0.01s

-1

0.001s

-1

20

20 0.0

-1

40

40

0.2

0.4

0.6

0.8

0.0

0.2

True strain

0.4

0.6

0.8

True strain

(b)

(a) 75

45

True Stress (MPa)

60

True Stress (MPa)

True Stress (MPa)

140

45

30

15

0 0.0

30

0.1s 0.01s

0.4 True strain

0.6

0.8

-1

15

19 0.2

-1

0.001s 0 0.0

0.2

0.4

0.6

-1

0.8

(d)

(c)

Fig. 13 Comparisons between the experimental and predicted flow stresses by the modified J-C model at the temperatures of a) 950 °C；b) 1050 °C；c) 1150 °C；d) 1250 °C

Table 4 RMSE of modified J-C model(MJCM) and original J-C model(OJCM) at various temperatures and strain rates (MPa) T(°C) 

950 °C

1050 °C

1150 °C

1250 °C

OJCM

MJCM

OJCM

MJCM

OJCM

MJCM

OJCM

MJCM

0.1

13.26

5.02

15.75

4.34

15.50

6.61

13.81

5.03

0.01

18.72

8.37

12.90

3.45

11.17

8.05

7.78

4.29

0.001

18.32

6.93

11.41

4.61

9.37

3.52

12.04

5.90

6. Conclusions 1) The grain size after deformation is different at various temperatures and strain rates. There are mixed fine grains of lathy and irregular polygon at elevated temperature and high strain rate, while the shape of grains is irregular polygon at low temperature and strain rate. These results show that the temperature, strain rate and strain are the main influencing factors of deformation mechanism. And, the working hardening and softening depend on the coupling interaction of them during deformation processes. 2) The modified J-C model of the X12 was established, considering the coupling effects of strains, temperatures and strain rates on the behavior of material, which reflects the mechanisms of working hardening and softening. It has good predictability of the

20

behavior of material correlation between the predicted results and experimental datum as compared with the original model by applying the statistical method.

Acknowledgements The authors gratefully acknowledge the financial support from the Training Fund for Outstanding Young Teachers in Higher Education Institutions of Shanghai in China (Grant No. ZZGCD15034), the School Level Start-up Fund of Shanghai University of Engineering Science, China (Grant No. 2016-51) and the Innovative Activities of College Students of Shanghai University of Engineering Science, China (cx1605004) and National Natural Science Foundation of China (Grant No. 51705316).

References [1] A. Fujio, Research and development of heat resistant materials for advanced USC power plants with steam temperatures of 700℃ and above, Eng. 1 (2015) 211-224. http://dx.doi.org/10.15302/J-ENG-2015031. [2] H. R. Cui, F. Sun, K. Chen, L. Zhang, R. Wan, A. Shan, J. S. Wu, Precipitation behavior of laves phase in 10%Cr steel X12CrMoWVNbN.10.1.1 during short-term creep exposure, Mater. Sci. Eng., A 527 (2010) 7505-7509. http://dx.doi.org/ 10.1016/j.msea.2010.08.013. [3] G. Götz, W. Blum, Influence of thermal history on precipitation of hardening phases in tempered martensite 10%Cr-steel X12CrMoWVNbN.10.1.1, Mater. Sci. Eng., A 348 (2003) 201-207. http://dx.doi.org/10.1016/S0921-5093(02)00684-6. [4] L. Han, R. Chen, J. Gu, J. Pan, Behavior of austenite grain growth in X12CrMoWVNbN.10.1.1 ferrite heat resistant steel, Acta Metall. Sinica 45 (2009) 1446-1450. http://www.ams.org.cn/EN/Y2009/V45/I12/1446. [5] K. Rodak, A. Hernas, A. Kielbus, Substructure stability of highly alloyed martensitic steels for power industry, Mater. Chem. Phys. 81 (2003) 483-485. http://dx.doi.org/10.1016/S0254-0584(03)00047-6.

21

[6] X. Xiao, G. Q.Liu, B. F. Hu, X. Zheng, L. N. Wang, S. J. Chen, A. Ullah, A comparative study on Arrhenius-type constitutive equations and artificial neural network model to predict high-temperature deformation behaviour in 12Cr3WV steel, Comput.

Mater.

Sci.

62

(2012)

227-234.

http://dx.doi.org/10.1016/j.commatsci.2012.05.053. [7] J. Ding, G. Kang, Q. Kan, Y. Liu, Constitutive model for uniaxial time-dependent ratcheting of 6061-T6 Aluminum alloy, Comput. Mater. Sci. 57 (2012) 67-72. http://dx.doi.org/10.1016/j.commatsci.2011.06.015. [8] J. Cui, H. Yang, Z. Sun, H. Li, Z. Li, C. Shen, Flow behavior and constitutive model using piecewise function of strain for TC11 Alloy, Rare Met. Mater. Eng. 41 (2012) 397-401. http://dx.doi.org/10.1016/S1875-5372(12)60035-6. [9] C. L. Gan, K. H. Zheng, W. J. Qi, M. J. Wang, Constitutive equations for high temperature flow stress prediction of 6063 Al alloy considering compensation of strain, Trans.

Nonferrous

Met.

Soc.

China,

24

(2014)

3486-3491.

http://dx.doi.org/10.1016/S1003-6326(14)63492-0. [10] S. A. H. Kordkheili, M. M. Ashrafian, H. Toozandehjani, A rate-dependent constitutive equation for 5052 aluminum diaphragms, Mater. Des. 60 (2014) 13-20. http://dx.doi.org/10.1016/j.matdes.2014.03.003. [11] Y. Chen, A. H. Clausen, O. S. Hopperstad, M. Langseth, Stress-strain behaviour of aluminium alloys at a wide range of strain rates, Int. J. Solids Struct. 46 (2009) 38253835. http://dx.doi.org/10.1016/j.ijsolstr.2009.07.013. [12] P. J. Zerilli, R. W. Armstrong, Dislocation-mechanics-based constitutive relations for material dynamics calculations, J. Appl. Phys. 61 (1987) 1816-1825. http://dx.doi.org/10.1063/1.338024. [13] D. L. Preston, D. L. Tonks, D. C. Wallace, Model of plastic deformation for extreme

loading

conditions,

J.

Appl.

Phys.

93

(2003)

211-220.

http://dx.doi.org/10.1063/1.1524706. [14] S.Wolfram, Cellular automata as models of complexity, Nat. 311 (1984) 419-424. http://dx.doi.org/10.1038/311419a0. [15] T. Mirzaie, H. Mirzadeh, J. M. Cabrera, A simple Zerilli-Armstrong constitative equation for modeling and prediction of hot deformation flow stress of steels, Mech. Mater. 94 (2016) 38-45. http://dx.doi.org/10.1016/j.mechmat.2015.11.013.

22

[16] R. Zhu, Q. Liu, J. Li, S. Xiang, Y. Chen, X Zhang, Dynamic restoration mechanism and physically based constitutive model of 2050 Al-Li alloy during hot compression,

J.

Alloys

Compd.

650

(2015)

75-85.

https://doi.org/10.1016/j.jallcom.2015.07.182. [17] L. X. Wang, G. Fang, M. A. Leeflang, J. Duszczyk, J. Zhou, Constitutive behavior and microstructure evolution of the as-extruded AE21 magnesium alloy during hot compression

testing,

J.

Alloys

Compd.

622

(2015)

121-129.

http://dx.doi.org/10.1016/j.jallcom.2014.10.006. [18] G. R. Johnson, W. H. Cook, Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures, Eng. Fract. Mech. 21 (1985) 31-48. http://dx.doi.org/10.1016/0013-7944(85)90052-9. [19] C. Zener, H. Hollomon, Effect of strain rate upon plastic flow of steel, J. Appl. Phys. 15 (1944) 22-32. http://dx.doi.org/10.1063/1.1707363. [20] A. Rusinek, J. A. Rodríguez-Martínez, A. Arias, A thermo-viscoplastic constitutive model for FCC metals with application to OFHC copper, Int. J. Mech. Sci. 52 (2010) 120-135. http://dx.doi.org/10.1016/j.ijmecsci.2009.07.001. [21] P. Li, K. Xue, Y. Lu, J. Tan, Neural network prediction of flow stress of Ti-15-3 alloy under hot compression, J. Mater. Process. Tech. 148 (2004) 235-238. http://dx.doi.org/10.1016/j.jmatprotec.2003.07.013. [22] S. Salari, M. Naderi, W. Bleck, Constitutive modeling during simultaneous forming and quenching of a boron bearing steel at high temperatures, J. Mater. Eng. Preform. 24 (2015) 808-815. http://dx.doi.org/10.1007/s11665-014-1345-5. [23] Z. Akbari, H. Mirzadeh, J. Mcabrera, A simple constitutive model for predicting flow stress of medium carbon microalloyed steel during hot deformation, Mater. Des. 77 (2015) 126-131. http://dx.doi.org/10.1016/j.matdes.2015.04.005. [24] Y. C. Lin, L. T. Li, Y. Q. Jiang, A phenomenological constitutive model for describing thermo-viscoplastic behavior of Al-ZnMg-Cu alloy under hot working condition, Exp. Mech. 52 (2012) 993-1002. http://dx.doi.org/10.1007/s11340-011-95464. [25] J. Q. Tan, M. Zhan, S. Liu, T. Huang, J. Guo, H Yang, A modified Johnson-cook model for tensile flow behaviors of 7075-T7451 aluminum alloy at high strain rates, Mater. Sci. Eng., A 631 (2015) 214-219. https://doi.org/10.1016/j.msea.2015.02.010.

23

[26] F. Ducobu, E. Rivière-Lorphèvre, E. Filippi, Constitutive model and their influence on the results of a Ti6Al4V orthogonal cutting model, Inter. J. Mech. Sci. 122 (2017): 143-155. https://doi.org/10.1016/j.ijmecsci.2017.01.004. [27] A. He, G. Xie, H. Zhang, X. Wang, A modified Zerilli-Armstrong constitutive model to predict hot deformation behavior of 20CrMo alloy steel, Mater. Des. 56 (2014) 122-127. http://dx.doi.org/10.1016/j.matdes.2013.10.080. [28] Y. C. Lin, X. M. Chen, G. Liu, A modified Johnson-Cook model for tensile behaviors of typical high-strength alloy steel, Mater. Sci. Eng., A 527 (2010) 69806986. http://dx.doi.org/10.1016/j.msea.2010.07.061. [29] W. Song, J. Ning, X. Mao, H. Tang, A modified Johnson-Cook model for titanium matrix composites reinforced with titanium carbide particles at elevated temperatures, Mater. Sci. Eng., A 576 (2013) 280-289. http://dx.doi.org/10.1016/j.msea.2013.04.014. [30] H. Y. Li, X. F. Wang, J. Y. Duan, J. J. Liu, A modified Johnson Cook model for elevated temperature flow behavior of T24 steel, Mater. Sci. Eng., A 577 (2013) 138146. http://dx.doi.org/10.1016/j.msea.2013.04.041. [31] Y. Zhu, Z. Zhu, Z. Xiang, Microstructural evolution in 4Cr(10)Si(2)Mo at the 4Cr(10)Si(2)Mo/Nimonic 80A weld jont by inertia friction welding, J. Alloys Compd. 476 (2009) 341-347. https://doi.org/10.1016/j.jallcom.2008.08.062. [32] T. Børvik, O. S. Hopperstad, T. Berstad, On the influence of stress triaxiality and strain rate on the behaviour of a structural steel―Part II: Numerical study, Eur. J. Mech. A−Solids 22 (2003) 15-32. http://dx.doi.org/10.1016/S0997-7538(02)00005-0. [33] A. H. Clausen, T. Børvik, O. S. Hopperstad, A Benallal, Flow and fracture characteristics of aluminium alloy AA5083-H116 as function of strain rate, temperature and

triaxiality,

Mater.

Sci.

Eng.,

A

364

(2004)

260-272.

http://dx.doi.org/10.1016/j.msea.2003.08.027. [34] Y. C. Lin, G. Liu, A new mathematical model for predicting flow stress of typical high-strength alloy steel at elevated high temperature, Comp. Mater. Sci. 48 (2010) 5458. http://dx.doi.org/10.1016/j.commatsci.2009.06.026. [35] D. Samantaray, S. Mandal, U. Borah, A. K. Bhaduri, P. V. Sivaprasad, A thermoviscoplastic constitutive model to predict elevated-temperature flow behaviour in a titanium-modified austenitic stainless steel, Mater. Sci. Eng., A 526 (2009) 1-6. http://dx.doi.org/10.1016/j.msea.2009.08.009.

24

[36] J. H. Hollomon, J. Member, Tensile deformation, Met. Tech., 12 (1945) 268-290.

25