1-s2.0-S0921509315002695-main

A physically-based constitutive model for SA508-III steel: Modelingand experimental verication

Dingqian Dong a, Fei Chen a,b,n, Zhenshan Cui a,n

a National Die & Mold CAD Engineering Research Center, Shanghai Jiao Tong University, 1954 Huashan Rd., Shanghai 200030, Chinab Department of Mechanical, Materials and Manufacturing Engineering, University of Nottingham, Nottingham NG7 2RD, UK

a r t i c l e i n f o

Article history:Received 28 September 2014Received in revised form10 March 2015Accepted 10 March 2015Available online 18 March 2015

Keywords:SA508-III steelWork hardeningDynamic recrystallizationFlow stressForging

a b s t r a c t

Due to its good toughness and high weldability, SA508-III steel has been widely used in the componentsmanufacturing of reactor pressure vessels (RPV) and steam generators (SG). In this study, the hotdeformation behaviors of SA508-III steel are investigated by isothermal hot compression tests withforming temperature of (9501250)1C and strain rate of (0.0010.1)s1, and the corresponding owstress curves are obtained. According to the experimental results, quantitative analysis of workhardening and dynamic softening behaviors is presented. The critical stress and critical strain forinitiation of dynamic recrystallization are calculated by setting the second derivative of the third orderpolynomial. Based on the classical stressdislocation relation and the kinetics of dynamic recrystalliza-tion, a two-stage constitutive model is developed to predict the ow stress of SA508-III steel.Comparisons between the predicted and measured ow stress indicate that the establishedphysically-based constitutive model can accurately characterize the hot deformations for the steel.Furthermore, a successful numerical simulation of the industrial upsetting process is carried out byimplementing the developed constitutive model into a commercial software, which evidences that thephysically-based constitutive model is practical and promising to promote industrial forging process fornuclear components.

& 2015 Elsevier B.V. All rights reserved.

1. Introduction

Knowledge of the high-temperature deformation behavior ofmetals and alloys is very important for the numerical modeling ofmany industrial processes. Prerequisite for a successful modeling ofhot working processes by means of numerical techniques, such asnite element and nite difference methods, one of the mostimportant items is a precise establishment of the constitutive equa-tions which describe the dependence of the ow stress on strain,strain rate and temperature. Therefore, an accurate constitutiveequation for the work-hardening and softening behavior is essential.Meanwhile, the understanding of ow behavior of metals and alloys athigh temperature is of great importance for designers for hot formingprocess, such as hot forging, rolling and extrusion. In the last twodecades, considerable investigations have been attempted to developconstitutive equations of materials from the experimentally measureddata to describe the hot deformation behavior [15].

RPV and SG are the key components of nuclear power equip-ment which dominate the lifespan of nuclear power plants. Atpresent, ASME SA508-III steel has been extensively used as thematerials of RPV and SG due to its high strength and toughness toprevent failure under severe working conditions [6]. As a keyequipment of the ultra-super-critical generator set, the RPV and SGshould have good mechanical properties through hot forging. Inthe past, many investigations have been carried out on thebehavior of SA508-III steel. Kim et al. investigated the failurebehaviors of the weld heat-affected zones, strain aging and fatiguecrack propagation under certain experiment conditions [7]. At thesame time, a new heat treatment process was developed toimprove the toughness for SA508-III steel by Kim et al. [8]. Leeet al. studied the relationship of the composition, structure andmechanical properties for the steel [9]. Liu et al. studied thefracture toughness of SA508-III steel in the temperature rangefrom room temperature to 320 1C using the J-integral method [10].Based on the literature reviewed above, it is found that theprevious studies mainly focus on the service performance ofSA508-III steel. However, there are only few reports on theproperties of SA508-III steel during hot forming process. Therefore,in order to study the workability and to optimize the hot forgingprocessing parameters, it is highly necessary to investigate the

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/msea

Materials Science & Engineering A

http://dx.doi.org/10.1016/j.msea.2015.03.0360921-5093/& 2015 Elsevier B.V. All rights reserved.

n Corresponding authors at: National Die & Mold CAD Engineering ResearchCenter, Shanghai Jiao Tong University, 1954 Huashan Rd., Shanghai 200030, China.

E-mail addresses: [email protected], [email protected] (F. Chen),[email protected] (Z. Cui).

Materials Science & Engineering A 634 (2015) 103115

effects of different thermomechanical parameters on the owbehaviors for this steel.

This paper mainly focuses on the high-temperature deforma-tion behavior of the steel during hot compression at the tempera-tures from 950 1C to 1250 1C and strain rates from 0.001 s1 to0.1 s1 using Gleeble-3500 thermo-mechanical simulator. Basedon the experimental results, the ow stress constitutive equationsare established by introducing the classical stressdislocationrelation and the kinematical recrystallization theory. Then throughsecondary development of the subroutine, the developed owstress model is integrated into commercial software DEFORM3D.To verify the validity and effectiveness of the developed program,rstly the isothermal compression tests are simulated to predictthe loading force under experimental conditions, and then a three-dimensional hot forging process in the industrial productionenvironment is performed to compare the loading force obtainedby the numerical simulation by using the developed const-itutive model.

2. Experimental procedures

In this investigation, the chemical compositions (wt%) of thecommercial SA508-III steel are summarized in Table 1. Cylindricalspecimens were machined with a diameter of 10 mm and a heightof 15 mm. The hot compression tests were carried out on aGleeble-3500 thermo-mechanical simulator according to the sche-dule illustrated in Fig. 1. The specimens were heated to 1250 1C at

Table 1The chemical composition of SA508-3 forging billet (wt%).

C Si Mn Cr Mo Ni Cu S P0.18 0.17 1.4 0.14 0.51 0.79 0.04 0.003 0.005

V Al N Co As Sn O H Sb0.005 0.022 0.008 0.0008 0.004 0.0042 0.006 0.0003 0.00005

Time (s)

Tem

pera

ture

(C

)

10C/s

1250C, 300s

10C/s

30sDeformation

Water quenching

Deformation temperature:950-1250C with 100C intervalStrain rate:0.001,0.005,0.01,0.1/sStrain:0.6

Fig. 1. Schematic representation of hot deformation process used in the experiments.

0.0 0.1 0.2 0.3 0.4 0.5 0.60

20

40

60

80

100

120

140

160

True strain

950C

1250C1150C

1050C

True

str

ess(

MPa

)

0.0 0.1 0.2 0.3 0.4 0.5 0.60

20

40

60

80

100

120

140

160

True strain

1250C1150C

1050C950C

True

str

ess(

MPa

)

0.0 0.1 0.2 0.3 0.4 0.5 0.60

20

40

60

80

100

120

140

160

True strain

True

str

ess(

MPa

)

1250C1150C

1050C950C

0.0 0.1 0.2 0.3 0.4 0.5 0.60

20

40

60

80

100

120

140

160

1250C1150C1050C950C

True strain

True

str

ess

(MPa

)

Fig. 2. Flow stressstrain curves of the SA508-III steel in different temperatures at the same strain rates (a) 0.001 s1, (b) 0.005 s1, (c) 0.01 s1 and (d) 0.1 s1.

D. Dong et al. / Materials Science & Engineering A 634 (2015) 103115104

a heating rate of 10 1C/s and held for 5 min and then cooled to thetest temperature at the cooling rate of 10 1C/s. Then, the specimenswere held at the forming temperature for 30 s to get a uniformtemperature distribution. The tests were performed at 950 1C,1050 1C, 1150 1C and 1250 1C and strain rates of 0.001 s1,0.005 s1, 0.01 s1 and 0.1 s1, respectively. The true stressstraincurves were recorded automatically in the isothermal compressionprocess. All specimens were compressed to a true stain of 0.6 andthen instantly quenched into cold water in order to preserve thehot deformation microstructure. In order to minimize the frictionbetween the specimen and the die during hot compression, theat ends of the specimens were covered by a lubricant consistingof graphite powder and machine oil. Finally, the quenched speci-mens were sliced along the axial section. All specimens weresliced along the axial section and etched with (5 g saturated picricacid 4 g SDBS100 mL H2O) solution at 6575 1C, and thenetching time varied from 10 s to 60 s to reveal austenite grainboundaries. The optical micrographs were recorded. Microstruc-ture evolution was observed by using Axio Imager M2m (Zeiss).The grain size was determined by using the line intercept methodin two vertical directions as per ASTM E112-88 [11]. The distancebetween adjacent parallel lines was 10 m, and the total truelength of the measuring line exceeded 1000 mm. The averagegrain size characterized as equivalent circle diameter was

calculated by multiplying the intercept length by a factor of(4/)1/2. The grain size distribution was obtained by dividing thenumber of grains in a certain range of the average grain diameterby the total number of measured grains.

3. Experimental result and discussion

3.1. Flow stress behavior

As shown in Fig. 2, a series of typical true stresstrue straincurves are obtained by hot compression tests under differentstrain rates and temperatures. Obviously, the ow stress issignicantly inuenced by the forming temperature and strainrate. In the initial deformation stage, the stress increases rapidlydue to the inuence of work hardening (WH), then the increasingrate of curves decreases with the increase of strain for theoccurring of DRX or dynamic recovery (DRV) until reaching peakstress p. When the softening rate exceeds hardening rate, thestress gradually reduces to a steady state due to a new balancebetween softening and hardening. According to the characteristicsof these curves, it can be seen that the values of peak stress pand static ow stress decrease with the increases of deformationtemperature and decrease of strain rate. As shown in Fig. 2,

10 20 30 40 50 60 70-150

0

150

300

450

600

750

Wor

k ha

rden

ing

Rat

e (

)

True stress (MPa)

950C 1050C 1150C 1250C

15 30 45 60 75 90-150

0

150

300

450

600

750 950C 1050C 1150C 1250C

True stress (MPa)

Wor

k ha

rden

ing

Rat

e(

)

20 30 40 50 60 70 80 90 100-100

0

100

200

300

400

500 950C 1050C 1150C 1250C

Wor

k ha

rden

ing

Rat

e (

)

True stress (MPa)20 40 60 80 100 120 140

-100

100

300

500

700

900 950C 1050C 1150C 1250C

True stress (MPa)

Wor

k ha

rden

ing

Rat

e(

)

Fig. 3. Work hardening rate versus true stress on the strain rate (a) 0.001 s1, (b) 0.005 s1, (c) 0.01 s1 and (d) 0.1 s1.

D. Dong et al. / Materials Science & Engineering A 634 (2015) 103115 105

majority of the curves show a single peak stress followed by agradual fall to a steady state stress which indicates that the typicalDRX behavior occurs with the increase of strain.

At a given strain rate, the ow stress and the peak strain (p)clearly drop with the increase of the deformation temperaturesrange from 950 1C to 1250 1C, respectively. However, when defor-mation temperature remains the same value, p and p willprogressively increase when the strain rate increases. At tempera-tures lower than 950 1C and strain rates higher than 0.1 s1, p isnot so apparent. When the deformation test is carried out at1250 1C within the range of the strain rates from 0.001 s1 to0.1 s1, p always appears and the ow stress curves fall fullyunder the type of DRX.

3.2. Critical strain for DRX

During hot deformation process, it is generally accepted thatonly when strain reaches a critical value can DRX occur [12]. Thevalues of work hardening rate are obtained by using thefollowing equation [1315]:

j _; T : i1i1i1i1

j _; T : 1

where strain rate _ and temperature (T) are constants in eachow curve. By using the method proposed by Poliak and Jonas[13], the critical stress c for initiation of DRX were determined.

The inection point is detected by tting a third order polynomialto the curves up to the peak point.

A13A22A3A4 2

3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0-8

-7

-6

-5

-4

-3

-2

-1

T=950C T=1050C T=1150C T=1250C

lnp

ln

0 20 40 60 80 100 120 140-8

-7

-6

-5

-4

-3

-2

-1

T=950C T=1050C T=1150C T=1250C

p(MPa)

ln

-1.0 -0.5 0.0 0.5 1.0 1.5 2.0-8

-7

-6

-5

-4

-3

-2

-1

T=950C T=1050C T=1150C T=1250C

ln(sinh(p))

ln

6.5x10-4 7.0x10-4 7.5x10-4 8.0x10-4 8.5x10-4-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

ln(s

inh(

p))

0.001s-1

0.005s-10.01s-1

0.1s-1

1/T

Fig. 4. Plots used for calculation of hot deformation constants (a) , (b) n0 , (c) n, and (d) value of slope.

-1.0 -0.6 -0.2 0.2 0.6 1.0 1.4 1.822

24

26

28

30

32

34

36

T=950C T=1050C T=1150C T=1250C

lnZ

ln(sinh(p))

Fig. 5. Plot used regression determinate the relationship ln Z versus ln sinhp.


where A1, A2, A3 and A4 are constants for a given deformationconditions. Differentiation of Eq. (2) with respect to results in:

0 dd

3A122A2A3 3

d2

d2 6A12A2 4

Then the minimum point of the second derivative of Eq. (4)corresponds to the critical stress c. When equals to zero, andthe critical stress can be expressed as

c A23A1

5

Then the critical work hardening c can be obtained at thecritical stress point from Eq. (2) as following equation:

c 2A32

27A21A2A3

3A1A4 6

0c can be expressed as

0c A3A223A1

7

22 24 26 28 30 32 34 36-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

T=950C T=1050C T=1150C T=1250C

lnZ

ln p

22 24 26 28 30 32 34-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

T=950C T=1050C T=1150C T=1250C

lnZ

ln

c

22 24 26 28 30 32 34 36-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

T=950C T=1050C T=1150C T=1250C

s

atln

(sin

h(

))

22 24 26 28 30 32 34 36-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

T=950C T=1050C T=1150C T=1250C

lnZlnZ

s

sln

(sin

h(

)

)

Fig. 6. Plot used regression determinate the relationship ln p (a), ln c (b), ln sinhsat (c), and ln sinhss (d) versus ln Z.

21 23 25 27 29 31 33 352.7

2.9

3.1

3.3

3.5

3.7

3.9

T=950C T=1050C T=1150C T=1250C

lnZ

ln

0

Fig. 7. Relationship between ln 0 and ln Z.


It is obvious that the saturation stress s is the intersection ofthe tangent extended curve with the line which corresponds to 0. And then the saturation stress expression can be obtainedusing linear regression.

2A32

27A21A2A3

3A1A4

! A3

A223A1

! A2

3A1

8

s 9A1A2A32A3227A21A4

27A21A39A1A22 A23A1

9

According to this method, the critical points for the initiation ofDRX can be obtained from the derivative of the third orderpolynomial versus true stress. Based on the ow curves, the valuesof critical strain were determined. The relationship between and

at the same strain rate and different temperatures are illustratedin Fig. 3. Moreover, the values of characteristic points such as peakstress p, steady-state stress ss and saturation stress s arealso obtained as shown in Fig. 3.

3.3. The deformation parameters of determination from experimentresults

In hot deformation of metallic materials, the Arrhenius equa-tions are widely used to describe the relationship of the strainrates, temperature and ow stress [1618]:

_ AFexp QactRT

10

Fig. 8. Microstructure of tested steel at strain rate0.1 s1on different deformation temperature (a) 950 1C, (b) 1050 1C, (c) 1150 1C and (d) 1250 1C; (e) strain rate0.01 s1and (f) strain rate0.001 s1 at deformation temperature 1150 1C.


where F is a function of ow stress with the followingequations:

F n0 o0:8exp Z1:2 sinhn for all

8>: 11

where _ is strain rate, R is the gas constant (8.314 J mol1 K1), T isthe absolute temperature. Qact is the activation energy of deforma-tion. is the ow stress, and ,,n0, n are the constants =n0.The effects of the temperature, strain and strain rate on thedeformation behaviors can also be formulated by ZenerHollomonparameter Z. The change of Z parameter is in accordance with thevariety of thep. The relationship between p and Z is usuallyexpressed by following equation [1921]:

Z _ exp QRT

A sinhp

n 12In order to determine the constant parameters, p was used as

the input data for the linear regression process. Substitution ofF from Eqs. (10)(12) and taking the natural logarithms for bothsides of the resulting equations, partial differentiation of theseequations yields to the following equations:

n0 ln _ ln p

T constant

13

ln _p

T constant

14

n ln _ ln sinhp

" #

T constant15

As shown in Fig. 4a and b, the slope of ln _ versus ln p and theslope of ln _ versus p are obtained. The average values are 6.6387and 0.1261 for n0 and , respectively. The value of =n0

is 0.019.

There is a linear relationship between ln sinhp

and ln _ whenT remains constant. In Fig. 4c, the value of n can be calculated fromthe relationship of ln _ and ln sinhp

, and the average value of

n is 4.795.The hot deformation Qact can be regarded as a signicant index

of the deformation difculty degree. The Qact value of SA508-IIIsteel can be expressed as following:

Qact Rn ln sinh p

1=T

_ constant

16

According to the relationship of ln sinhp

versus 1=T , thelinear slope increases with the increase of strain rate, the averagevalue of these slopes is 9434.037 as shown in Fig. 4d. According toEq. (16), the Qact of SA508-III steel during hot deformation can beobtained as 376,088 J/mol.

Fig. 5 illustrates the relationship between ln Z and ln sinhp

.Eq. (17) can be obtained by taking the logarithm of both sides of Eq.(12).

ln Z n ln sinhp ln A 17

By regression from the resultant equation in Fig. 5, a newequation was obtained as following:

Z _ exp 376:088 103

RT

! 8:5983 1011

sinh0:019p 4:6899 18

In order to establish the expression of feature parameters, thevalues of characteristic points such as peak strain p, criticalstrain c, saturation stress s and steady-state stress ss can beeasily obtained from the ow stress curves. Fig. 6 ad shows thatthe linear relation exist between ln p, ln c, lnsinhs, and

lnsinhss versus ln Z, respectively. So, by tting the result ofthe experiment in Fig. 6, p, c, s and ss can be expressed as afunction of the ZenerHollomon parameter (Z), respectively.

p 1:384 103Z0:1715 19

c 0:0004 Z0:19602 20

s 52:687 sinh1 0:003646Z0:2092

21

ss 52:687 sinh1 0:002805Z0:2070

22

The initial stress 0 at various forming temperatures andstrain rates can be directly obtained from the true stressstraincurves, and in this study it is taken as the initial stress 0 0:2%.Fig. 7 illustrates that a good linear relationship exists betweenln 0 and ln Z. Then the initial stress 0 can be expressed as afunction of ZenerHollomon parameter (Z),

0 2:1829Z0:08901 23

22 24 26 28 30 32 34 360

2

4

6

8

10

12

14

16

18

20

22

lnZ

lnk i

lnk1 lnk2k1 value Line Fit

k2 value Line Fit

Fig. 9. Plot used regression determinate the relationship between ln ki and ln Z.

-3 -2 -1 0 1 2 3-6-5-4-3-2-10123456

T=1050C,0.001sT=1150C,0.001sT=1250C,0.001sT=1050C,0.005sT=1150C,0.005sT=1250C,0.005s

T=1050C,0.01sT=1150C,0.01sT=1250C,0.01sT=1050C,0.1sT=1150C,0.1sT=1250C,0.1s

ln(-l

n(1-X d

rx))

ln((c)/p)

Fig. 10. Linear t value of kd and nd by regressed DRX parameters.


4. Model establishment for dynamic recrystallization

4.1. DRX microstructures

The microstructural evolution of hot deformed SA508-III steelwere investigated at a strain of 0.6 under various formingtemperatures and strain rates. Fig. 8ad illustrates the opticalmicrostructures after deformation at strain rate of 0.1 s1

under the temperatures of 950 1C, 1050 1C, 1150 1C and 1250 1C,respectively. It is obvious that the deformed austenite grain sizeincreases with the increase of the forming temperature. Theaverage grain sizes were measured as 35 m, 53 m, 81 m and117 m for the forming temperatures of 950 1C, 1050 1C, 1150 1Cand 1250 1C, respectively. The measured grain size results agreereasonable with the characteristic of the ow stress curve. Asshown in Fig. 2d, it can be seen that the ow stress peak is not soapparent at strain rate 0.1 s1 at 950 1C, and the ow stress curveexhibits the characteristics of dynamic recovery, and then thedynamic recrystallization does not occur. However, it can beobserved that the dynamical recrystallization occurs with theincrease of the forming temperature. The main reason for thisphenomenon is that higher forming temperatures lead to theincrease of deformation stored energy. And then the dynamical

recrystallization grain size is rened with decreasing the deforma-tion temperature.

Fig. 8c, e and f illustrates the optical microstructures of teststeel after deformation at 1150 1C under strain rates of 0.1 s1,0.01 s1 and 0.001 s1, respectively. It can be easily found that thehigher the strain rates, the ner the grain sizes, which weremeasured as 81 m, 113 m and 144 m under the strain ratesof 0.1 s1, 0.01 s1 and 0.001 s1, respectively. It can be clearlyseen that the recrystallized grain size becomes ner with theincrease of the strain rate. This phenomenon is mainly attributedto two reasons [12]: higher strain rate increases the work hard-ening rate and reduces the time for austenite recrystallization,which leads to insufcient time for austenite grain to grow up at acertain strain, in other words the recrystallized nucleus under ahigher strain rate has insufcient time to fully grow up.

4.2. Mathematical Model for DRX

4.2.1. Kinetics of DRXDRX is an important phenomenon for controlling microstruc-

ture and mechanical properties in hot working. The onset of DRXmainly depends on the distribution and density of dislocationduring deformation. Based on the classical stressdislocation

0.0 0.1 0.2 0.3 0.4 0.5 0.60

20

40

60

80

100 0.1s-1

0.01s-1

0.005s-1

0.001s-1

Volu

me

frac

tion

of D

RX,

Xdrx

Strain

0.0 0.1 0.2 0.3 0.4 0.5 0.60

20

40

60

80

100 0.1s-1

0.01s-1

0.005s-1

0.001s-1

Volu

me

frac

tion

of D

RX,

Xdrx

Strain

0.0 0.1 0.2 0.3 0.4 0.5 0.60

20

40

60

80

100 0.1s-1

0.01s-1

0.005s-1

0.001s-1

Volu

me

frac

tion

of D

RX,

Xdrx

Strain0.0 0.1 0.2 0.3 0.4 0.5 0.6

0

20

40

60

80

100

Volu

me

frac

tion

of D

RX,

Xdrx

Strain

0.1s-1

0.01s-1

0.005s-1

0.001s-1

Fig. 11. Dynamic recrystallization fraction for SA508- steel under the different strain rates with deformation temperature: (a) 950 1C, (b) 1050 1C, (c) 1150 1C and(d) 1250 1C.


relation and the kinematical recrystallization proposed by Laas-raoui and Jonas [22,23], the ow stress constitutive equations ofthe WH-DRV period and DRX period were established, respec-tively. According to effects of the WH and DRV, the evolution ofthe dislocation density with strain is generally considered todepend on the following two components [2426]:

dd

k1

p k2 24

where d=d is the rate of increase of dislocation density withstrain, k1 represents the coefcient of WH, and k2 is the coefcientof DRV. When 0, 0, where 0 is the initial dislocationdensity.

k1k2k1k2e

k22 0p e k22

225

when

dd

0; s k1k2

226

s is the saturation dislocation density corresponding to thesaturation stress s. Previous studies have shown that the effectivestress is negligible compared to the internal stress at high

temperature and the mechanical strength of the obstacles todislocation glide, so the applied stress can be related directly to

0.0 0.1 0.2 0.3 0.4 0.5 0.60

20

40

60

80

100

120

140

160

1250C1150C

1050C950C

Predicted Experimental

True

str

ess(

MPa

)

True strain

Strain rate:0.001s-1

0.0 0.1 0.2 0.3 0.4 0.5 0.60

20

40

60

80

100

120

140

160

1250C1150C

1050C

950C

Strain rate:0.005s-1

True strain

True

str

ess(

MPa

)


0.0 0.1 0.2 0.3 0.4 0.5 0.60

20

40

60

80

100

120

140

160

1250C1150C

1050C950C

Strain rate:0.01s-1

True

str

ess(

MPa

)


True strain0.0 0.1 0.2 0.3 0.4 0.5 0.60

20

40

60

80

100

120

140

160

1250C1150C1050C950CStrain rate:0.1s-1

True

str

ess

(MPa

)

True strain


Fig. 12. Comparisons experimental and predicted ow stress value of SA508- steel under various hot deformation conditions.

Fig. 13. Correlation between the experimental and predicted ow stress data fromthe developed constitutive equation.


the square root of the dislocation density [14]. The structuralparameter can be selected as b p . is the materialconstant, is the shear modulus and b is the distance betweenatoms in the slip direction [27]. Therefore, considering the impactof WH and DRV during hot deformation, the ow stress as afunction of strain can be given as the following expression.

WH s0sek22 ; oc 27

According to the strain and stress determined by the experi-ment ow stress curves, k2 can be calculated under deformationconditions. Therefore, Eq. (27) is derived as

k2 2ln

0sWHs

28

k1 can also be calculated from Eq. (26) and b p .Considering the shear modulus changes with deformationtemperature, the equation [28] can be expressed as

k1 1

0 10:91 T300Tm

0@

1Ask2 29

where 0 is the shear modulus when temperature is 300 K, Tm ismelting point. Therefore, regression model can be established inthe Eq. (24). A1 and n1 are constants, so the value of k1, k2 can bedetermined in Fig. 9.

k2 A1Zn1 30

Regression equation is obtained as follows:

k2 319:1847Z0:0788 31

At high temperatures and low strain rates the DRX phenom-enon is more obvious. The volume fraction of DRX (Xdrx) can bedetermined by [2527,29]:

Xdrx 1exp kdrxcP

ndrx ; Zc 32

Meanwhile, the progress of the DRX, Xdrx, can also be written as

Xdrx WHsss

; Zc 33

By combining Eqs. (28) and (29), the ow stress during DRXperiod can be given by the following expression:

WHsss 1exp kdrxcp

ndrx

; Zc 34

where Xdrx is the volume fraction of DRX. kdrx and ndrx are DRXparameters depending on chemical composition and hot deforma-tion conditions.

Then, Eq. (32) can be rewritten as following,

ln ln1Xdrx ln kdrxndrx lncP

35

The relationship between ln ln1Xdrx

and lnc=pis used to determine kdrx and ndrx. As shown in Fig. 10, the kineticmodel of DRX can be acquired with the average value of kdrx and

Start

Input data

t=t+t

Read simulation results

Strain increment

Output data

End

YesNo

No

Yes

Fig. 14. Flowchart of the secondary development of the ow stress subroutine.

0.0 1.5 3.0 4.5 6.0 7.50

5000

10000

15000

20000

25000

Stroke,mm

Load

For

ce, N

Strain rate:0.1s-1

Experiment Simulation

950C

1050C

1150C

1250C

Fig. 15. Variation of anvil load force with stroke for SA508-III steel under strainrates 0.1 s1with deformation temperature: (a) 950 1C, (b) 1050 1C, (c) 1150 1C and(d) 1250 1C.


ndrx which are 0.6949 and 1.7916, respectively.

Xdrx 1exp 0:6949cP

1:7916 !36

Fig. 11 shows the dynamic recrystallization volume fraction ofSA508-III steel calculated by Eq. (36) under different strain ratesand temperatures. As shown in the gures, the volume fraction ofDRX increases with the increasing of the strain. For a certain strain,the volume fraction of DRX is higher at higher deformationtemperature or lower strain rate. Therefore, under the formationconditions such as higher deformation temperatures and lowerstrain rates, DRX tends to be complete; in other words, the volumefraction of DRX is close to 100%.

Therefore, the physical-based constitutive relation of SA508-IIIsteel during the WH-DRV period and DRX period can be

summarized as

WH s0sek22 ; oc

WHsss 1exp kdrxcp ndrx

h in o; Zc

0 2:1829Z0:08901c 0:0004 Z0:19602p 0:001384 Z0:17152;s 52:687 sinh1 0:003646Z0:2092

ss 52:687 sinh1 0:002805Z0:2070

k2 319:1847Z 0:0788kdrx 0:6949;ndrx 1:7916Z _ exp 376088RT

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

37

4.2.2. Verication of the developed constitutive equationIn order to verify the developed constitutive models, compar-

isons between the experimental data and predictions have beenperformed in Fig. 12ad. It can be easily found that there is a goodagreement between the predicted value and the experimentaldata. It illustrates that the proposed constitutive equation gives areasonable estimate of the ow stress for this steel.

In order to further evaluate the prediction accuracy of thedeveloped constitutive model, the predictability of the developedequation can be quantied in terms of standard statistical para-meters such as correlation coefcient (R) and the average absoluterelative error (AARE) as shown in Fig. 13. The AARE is calculated bycomparing the relative errors and therefore is an unbiasedstatistical parameter for determining the predictability of theequation. The correlation coefcient (R) provides information on

Fig. 16. The forming process of experiment (a) preheating of the die, (b) initial forging state, (c) nal 80% reduction forming and (d) dimension measurement.

Table 2Simulation conditions for the hot upsetting experiments.

Process condition Value of parameter

Initial temperature of workpiece 1200 1CInitial temperature of dies 300 1CMaterial of dies H13Number of workpiece units 56018Number of Up-die units 16958Number of Bottom-die units 16866Thermal conductivity of environment 0.02 N/s/mm/1CThermal conductivity between die and workpiece 5 N/s/mm/oCFriction factor 0.3Upper die velocity 2 mm/sIncrement of time 0.1 s/stepIncrement of step 0.5 mm/step


the strength of linear relationship between the experimental andthe predicted values. These are expressed as following equations:

RPN

i 1XiXYiYPNi 1 XiX2

q PNi 1 YiY2

q 38

AARE 1N

XNi 1

YiXiXi

100% 39

where Xi, Yi are the measured and predicted ow stress, respec-tively. X, Y are the mean values of Xi and Yi, respectively. N is thetotal number of data used in this investigation. The calculatedcorrelation coefcient (R) is 0.997, which indicates that there is agood correlation between the predicted and measured data.Meanwhile, the average absolute relative error (AARE) is only3.26%, which illustrates the good prediction capability of thedeveloped model. Therefore, the developed constitutive equationcan be applied to numerical simulation for describing the owbehavior of the SA508- steel under the elevated formingtemperature.

4.3. Experimental verication

In order to carry out numerical simulation of hot forging, thedeveloped consitituve euqations were implemented into threedimensional DEFORM commercial software. Fig. 14 gives theowchart of secondary development of the ow stress subroutine.

To verify the validity and effectiveness of the developed program,the nite element method simulations were carried out tocalculate the loading force under different experimental condi-tions. Fig. 15 shows the variation of loading force with strokeunder four different deformation temperatures, 950 1C, 1050 1C,1150 1C and 1250 1C at the strain rate of 0.1 s1. As shown in thegure, the maximum relative error is about 5.44% with deforma-tion temperature 950 1C at strain 0.6. Generally speaking itillustrates that the loading force simulated by FEM ts in wellwith the results of compression test. In other words, the simula-tion results conrm that the developed constitutive equations welldescribe the ow behavior of SA508-III steel during hot formingprocess.

Further, a three-dimensional hot upsetting experiment in theindustrial production environment was performed. The cylinderworkpiece with a diameter of 40 mm and a height of 60 mm wasused in the experiment. The workpiece was heated to 1200 1C andheld for 300 min to obtain a homogeneous initial temperature byusing the vacuum furnace. The 320 ton capacity hydraulic presswas adopted in the experiment and the experiment was carriedout at a ram speed of 2 mms1 till 80% of reduction was reached.Fig. 16ad illustrates the whole hot forging processes includingpreheating of the die, initial forging state, nal forging state anddimension measurement. The experimental conditions and FEMsimulation parameters for the upsetting process are listed inTable 2. Fig. 17a and b shows the strain and stress elds obtainedby using DEFORM3D. The comparison between the load of theupdie during FEM simulation and the experiment load is shown inFig. 18. Clearly, there is a good agreement between the resultsobtained by experiment and FEM simulation. It demonstrates thatthe proposed deformation constitutive equations can be used toanalyze the numerical simulation during hot forging process forSA508- steel in the industrial production environment.

5. Conclusions

In this study, the hot deformation behavior of SA508-III steelhas been investigated by means of the compression test over apractical range of temperatures and strain rates. The ow stressconstitutive equations for the work hardening-dynamical recoveryperiod and dynamical recrystallization period were developed forSA508-III steel. By comparing the predicted and measured results,it is conrmed that the proposed constitutive equations can beused in numerical simulation in hot forming process of SA508-IIIsteel. By integrating the developed ow stress model into DEFO-RM3D, a three-dimensional hot upsetting experiment in the

Fig. 17. The results of numerical simulation (a) strain-effective and (b) stress-effective.

0 10 20 30 40 500

200000

400000

600000

800000

1000000

1200000 Experiment Simulation

Stroke,mm

Load

For

ce, N

Fig. 18. Comparison between the load of FEM simulation and the experimentalresults.


industrial production environment was simulated. It indicates thatthe developed numerical simulation platform is very effective tobe used in heavy forging industrial application for SA508-III steel.

Acknowledgments

This work was nancially supported by National Basic ResearchProgram of China (Grant no. 2011CB012903) and National Scienceand Technology Major Project (Grant no. 2011ZX04014-051).

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A physically-based constitutive model for SA508-III steel: Modeling and experimental verificationIntroductionExperimental proceduresExperimental result and discussionFlow stress behaviorCritical strain for DRXThe deformation parameters of determination from experiment results

Model establishment for dynamic recrystallizationDRX microstructuresMathematical Model for DRXKinetics of DRXVerification of the developed constitutive equation

Experimental verification

ConclusionsAcknowledgmentsReferences

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