Journal of Materials Processing Technology 140 (2003) 84–89
Study on constitutive relation of AISI 4140 steel subject to large strain at elevated temperatures S.I. Kim a,∗ , Y. Lee b,1, S.M. Byon c,2 a
c
POSCO Technical Research Laboratories, Automotive Steel Research Group, Kwangyang, South Korea b POSCO Technical Research Laboratories, Plate and Rod Research Group, Pohang, South Korea POSCO Technical Research Laboratories, Instrumentation and Control Research Group, Pohang, South Korea
Abstract
We conducted hot torsion and compression tests of AISI 4140 steel (medium carbon and low alloy steel) to investigate the effect of deformation mode on constitutive relation. It was noted that shapes of the curves formed by the experimental data were different, although the overall stress levels were similar. To formulate constitutive relation of AISI 4140 steel, various constitutive equations were considered. Among them, Voce’s constitutive equation was selected as the model to possibly modify. Results showed that the modified constitutive equation could predict the flow stress of AISI 4140 steel accurately, in comparison with the results of the other constitutive models such as Misaka’s and Shida’s equations. It has been found out that the approach to obtain a constitutive equation applicable to large strain ranges was fruitful and this modified equation might have a potential to be used for hot strip rolling process where more precise calculation of stress decrement due to dynamic recrystallization is important. © 2003 Elsevier B.V. All rights reserved. Keywords: Constitutive Constitutive equation; Deformation mode; Compression test; Torsion Torsion test; AISI 4140 steel; Recrystallization
1. Introducti Introduction on
The analys analysis is proble problem m of metal metal formin forming g proces processs (hot (hot rolling, forging and extrusion) has been dependent on various parameters including constitutive relation, the shape of the workpiece and product, the shapes of the tools, friction, temperatur temperature, e, forming forming speed, speed, etc. In such parameters, parameters, the constitutive relation is one of the most important factors having an effect on solution accuracy. A number of research groups have attempted to develop consti constitut tutiv ivee equati equations ons of materi materials als and sugges suggested ted their their own formulations by putting the experimentally measured data data into into one single single equati equation. on. Misaka Misaka [1–3] and Shida Shida [4] proposed [4] proposed the models giving the stress of carbon steels as a functi function on of the strain strain,, strain strain rate, temper temperatu ature re and carbon carbon content. content. Shida’s model takes takes account account of the flow stress behavior of the steels in austenite, ferritic and in the two-phase regions. Voce Voce suggested an approximate equation of stress–stain curve considering the dynamic recrystalliza∗
Corresponding author. Tel.: +82-61-790-8806; 82-61-790-8806; fax: +82-61-790-8801.
E-mail addresses:
[email protected] (S.I. Kim),
[email protected] (Y. Lee),
[email protected] (S.M. Byon). 1 Tel.: +82-54-220-6058; 82-54-220-6058; fax: +82-54-220-6911. 2 Tel.: 82-54-220-6363; fax: +82-54-220-6914. +82-54-220-6363; 0924-0136/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0924-0136(03)00742-8
tion [5] [5].. Johnso Johnson n and Cook [6] developed developed a constituti constitutive ve equation which assumes that the dependence of the stress on the strain, strain rate and temperature can be multiplicatively decomposed into three separate functions that include five constants to be determined by experimental data obtained for a specific material. Laasraoui and Jonas [7] followed the disloc dislocati ation on theory theory to deriv derivee a formul formulaa that that descri describes bes the strain hardening behavior. When dynamic recrystallization occurs, the softening behavior may be expressed by the Avrami equation, which is used to present the softening portion of the flow curve. However, the rate constant of dynamic dynamic recrystall recrystallizati ization on was not possible possible to express express mathematically as a function of strain rate and temperature. There are many different types of test methods, such as hot torsion, compression, tension, cam plastometer and drop hammer for developing the constitutive equation of materials. Among them, hot torsion and compression tests have widely been used. Hot torsion test has been used for many researchers to formulate constitutive equation of materials subject to a large strain because it has a forte in simulating the multi-pass deformation, in comparison with axisymmetric compression test. In the case of compression test, high friction at the interface of material and stroke head results in barreling during test, as reduction, i.e., natural strain, becomes large. Thus, the compression test has a limitation in
S.I. Kim et al. / Journal Journal of Materials Materials Processing Processing Technol Technology ogy 140 (2003) 84–89
generating flow stress curve when the material undergoes large strain [8–10] strain [8–10].. In this this study study,, we presen presentt a consti constitut tutiv ivee equati equation on for AISI4140 that experiences large strain at elevated temperatures and subsequently recrystallization during deformation. The AISI 4140 steel is one of the representative medium carbon and low alloy steels that have been used for a wide utility. However, the advance of quality, which depends on the control of microstructure such as recrystallization mode and grain size during hot rolling was not improved as yet. In hot rollin rolling g proces process, s, grain grain refinem refinement ent can be obtain obtained ed by using the controlled rolling technique which derives recrystallization during rolling. In addition, for securing high dimension accuracy of final product we have to optimize roll gap control model during multi-pass continuous rolling and consequently we need constitutive equation of the steel subject to large strain at elevated temperature. We first acquired flow stress curves obtained from torsion and compression test under the same deformation conditions. We then presented the differences between torsion and compression test mode in terms of work hardening and strain rate sensitivity coefficient. We used Voce’s constitutive equation to possibly describe the flow stress curve obtained from hot torsion tests and obtained the constitutive equation appropriate for AISI4140 steel, by modifying the Voce’s equation. The validity and fruitfulness of the proposed equation was demonstrated when AISI4140 steel undergoes large strain at elevated temperatures.
85
cylinder shape and and its dimension is 15 mm length and 10 mm diameter.
3. Comparison Comparison of torsion and compression compression test
Fig. 1 1 shows the measured stress–strain curves obtained from from tors torsio ion n and and comp compre ress ssio ion n test testss for for AISI AISI 4140 4140 stee steell at difdifferent strain rates at a fixed temperatures, 900 and 1000 ◦ C. During compression test at lower strain rate, some noises in data acquisition were observed. In all cases the flow stress obtained from compression test is higher than that from torsion test, although the general shapes of the measured curves are similar. The differences between the stresses are approximately in the range of 10–20%. These differences might be attributable to the following reasons: compression test has the glowing frictional forces at the ram-specimen interface as the test progress, while this effect is minor in torsion test. In compression test, it is difficult to achieve constant strain rate and isothermal condition during test, whereas to control them in torsion test is accurate [8–10] accurate [8–10].. To understand better the difference between compression mode and torsion mode, we need to determine the rela-
2. Experimental procedures
AISI 4140 steel with an initial grain size of about 120 m was was used used as the the samp sample less for for hot hot comp compre ress ssio ion n and and tors torsio ion n test test.. The chemical composition of the steel is listed in Table 1. 1. The material was obtained in the form of square as-cast billet billet with a side length length of 160 mm. The torsion test specspecimen imenss with with a gaug gaugee sect sectio ion n of 20 mm leng length th and and 5 mm radius were machined. To investigate the effects of temperature and strain rate on the flow stress, compression and torsion tests were conducted in the temperature range of 900–1100 ◦ C and the strain strain rate range of 0.05–5 0.05–5 s−1 . For the calculation of the surface shear stress from the torque, Fields–Backofen equation [11] equation [11] was was used. The hot compression sion test test has been performe performed d by GLEEB GLEEBLE LE 3500 C. The specimen was heated up to the desired temperature using direct resistance heating system inside the furnace and remains for 3 min for uniform uniform distribution distribution of temperatur temperaturee across the specimen, and then the test started. The specimen is of a
Table 1 The chemical composition of AISI 4140 steel C
Mn
Si Si
Cr
Mo
P
S
Fe
0.04
0.67
0.21
0.97
0.15
0.045
0.030
Bal.
Fig. 1. Representative flow stress curves obtained from hot torsion and compre compressi ssion on test test of AISI AISI 4140 4140 steel steel under under vario various us strain strain rates rates at (a) 900 ◦ C and (b) 1000 ◦ C.
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Table 2 The experimental constants of AISI 4140 steel
4. Constitutiv Constitutivee equations
Test method
A
α
n
Q (kJ/mol)
Torsion Com Compres ressio sion
1.05 × 10 12 2.43 .43 × 10 12
0.0109 0.0086
4.86 5.43
318 315
tionship tionship among strain rate, temperature, temperature, and flow stress. For this purpose, we used the equation associated with the ArrheniusArrhenius-type type temperature temperature sensitiv sensitivity ity term proposed proposed by Sellars et al. [12] al. [12]..
Q Z = ε˙ exp − RT
n
= A(sinh ασ p )
(1)
where R is the gas constant and A and α, material dependent constants. The n and Q are temperature independent constant and activation activation energy for deformation, respectively. respectively. This This hyperb hyperboli olicc sine sine functi function, on, which which combin combines es with with the Arrhenius-type temperature sensitivity term, describes the behavior of the materials deformed at different temperatures and strain rates. A temperature independent constant such as n can be determined from the relationship between stress term and strain rate (ε˙ ). The activation energy, Q can then be calculated from the relationship between stress and reciprocal of absolute temperature (1/ T [1,13–15].. In this study, T) [1,13–15] the A , α , n and Q are represented in Table in Table 2. 2. From the relation of peak stress (σ p) with temperature and strain strain rate, rate, we can determ determine ine the strain strain rate rate sensit sensitiv ivity ity,, m, and and work hardening exponent, n, which are material parameter that is important in judging the formability of the material in the particular operation. The material parameters of the steel are tabulated in Table in Table 3. 3. Strain rate sensitivity and strain hardening exponent obtained from compression test were, respectively, 153 and 207% higher than those from hot torsion test. This may be attributed to barreling of specimen during compression test and consequently more deformation resistance occurs. Considering that tensile straining at the cylindrical surfaces stems the level of straining from progressing, we can easily explain the limitation of compression test. In addition, the data obtained from compression test does not exist in the large strain range, indicating that the test is impossible for simulating a severe deformation and accurate description of deformation behavior of material. Hence we use experimental data obtained from torsion test for constructing constitutive equation.
Some constitutive equations where the effect of deformation variables, such as strain, strain rate and temperature on the flow stress appears explicitly has been compared and analyzed. Among them, we have studied thoroughly Misaka’s, Shida’s and Voce’s constitutive equations. 4.1. Misaka’s Misaka’s equation equation
Misaka and Yoshimoto [2] have have utilized utilized the following following double-power constitutive equation to determine flow stress associated with the processing of steels 2 σ Misaka Misaka = 9.8exp(0.126 − 1.75[C] + 0.594[C]
+
2851 + 2968[C] − 1120[C]2 T + 273
Test method
m
n
Torsion Compression
0.112 0.171
0.468 0.968
εn ε˙ m
(2)
Application range of this formula is as follows; carbon content: ∼1.2%, temperature: 750–1200 ◦ C, reduction (natural strain): ∼50% and strain strain rate: 30–200 30–200 s−1 . The strain rate sensitivity, m and work hardening exponent, n in Table in Table 3 are 3 are used. Misaka’s equation was updated by including effects of solution-strengthening and dynamic recrystallization [3] recrystallization [3].. The updated Misaka’s constitutive equation is (σ Misaka Misaka )updated = (fσ Misaka Misaka )(1 − Xdym ) + Kσ ss ss Xdym
(3)
f = 0.835 + 0.51[Nb] + 0.098[Mn] + 0.128[Cr]0.8 0.3
+ 0.144[Mo]
+ 0.175[V] + 0.01[Ni]
(4)
(σ Misaka Misaka )updated indicates the flow stress of steels containing multiple alloying additions. σ ss ss is steady state stress, K = 1.14 is a parameter that converts flow stress to mean flow stress, and X dym dym is volume fraction of dynamic recrystallization [3] lization [3].. It might be useful for practical purpose but its mathematical base is weak. The factors for Mn, Nb, V and Ni are linear although the terms for Cr and Mo are nonlinear. Devadas et al. [16] al. [16] c compared ompared predicted flow stress data for a low alloy steel with measured data from a cam-plastometer. They showed that Misaka’s model overestimated the flow stress. 4.2. Shida’s Shida’s equation equation
Shida’s equation [4] [4] is based on experimental data obtained from compression type of high strain rate testing machines. Shida then expressed the flow stress of steels, σ , as a function of the equivalent carbon content (C ), ), the strain (ε), the strain rate (ε˙ ) and temperature (T ) as followings: (5)
σ = σ d (C,T)f w (ε)f r (ε) ε˙ )
Table 3 The material parameters of AISI 4140 steel
σ d = 0.28 exp exp
[K] = T [K]
5.0 T
−
[ ◦ C] + 273 T [ 1000
0.01 C + 0.05
(6) (7)
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f w (ε) = 1.3
ε
n
0.2
− 0.3
n = 0.41–0.07 C f r (ε) ε˙ ) =
ε
(8)
0.2
(9)
o
120
1000 C, 0.5/sec Compression
100
m
ε˙
140
(10)
10
m = (−0.019C + 0.126)T + (0.076C − 0.05)
(11)
where f w (ε) and f r (ε) ε˙ ) are functions dependent upon strain and strain rate, respectively. The formulation of Eq. of Eq. (5) is based on assumption that flow stress increases with the strain rate and strain increased. The range of validity of the formula is quite broad. This formula is applicable in the range of carbon content: 0.07–1.2%, temperature: 700–1200 ◦ C, strain: ∼0.7 and strain rate: ∼100s−1 .
) a P 80 M ( s s e 60 r t S
Torsion
40
Developed Eq. Shida's Eq. Misaka's Misaka's Eq.
20 0 0. 0
0.1
0.2
0.3
0.4
0 .5
Strain Fig. 2. Comparison of measured and predicted constitutive relations for AISI 4140 steel in the region of WH + DRV.
4.3. Voce’s equation
In contrast contrast to Misaka’s Misaka’s and Shida’s Shida’s equations equations,, Voce’s equation can describe the flow stress over the wide range of strains and strain rates. The equation can express the dynamic softening portion of the flow stress curve by using Avrami equation [1,5,7,13–15] equation [1,5,7,13–15].. In this study, we developed a constitutive equation by modifying Voce’s constitutive equations accounting for the dynamic recrystallization as well as the dynamic softening. During thermomechnical processing, the important metallurgical phenomena are work hardening (WH), dynamic recovery (DRV) and dynamic recrystallization (DRX). Thus, the flow stress curve can be bisected with the region of WH + DRV and the DRX region. For the evaluation of the WH and DRV region, the following expression was used [13–15]:: [13–15] m σ (WH+DRV DRV) = σ p [1 − exp(−Cε)]
(12)
The coefficient, C , and work hardening exponent, m, are dependent on the deformation conditions. The parameters, C and m are normally taken as being a constant, however, it is a function of the deformation conditions (strain rate, temperature temperature). ). Thus, Thus, the C and m are considered to be a function of dimensionless parameter, Z / A (Table ( Table 4). 4). ∗ ε ε σ σ Also, critical values, such as , c , p and ss ss are a function of Z in Table 4. 4. These are obtained from the Z / A, as listed in Table in-detail analysis of flow curves at various deformation conTable 4 The parameters of AISI 4140 steel Test method
Expression
Torsion
C = 6.45( Z / A)−0.13 m = 0.26 ( Z / A)0.148 m = 1.578 ( Z / A)−0.02 εc = 0.252 ( Z / A)0.158 ∗ = 0.528 ( Z / A)0.1675 ε 15.24ln ( Z / A) σ p = 82.74+15.24ln 72.6 12.07ln 12.07ln ( Z / A) σ ss = + ss
ditions. Method to get these parameters are well-subscribed in [13 in [13–15] 15].. Fig. Fi g. 2 shows hows the measur measured ed and predic predicted ted flow stres stresss curve curvess in the region of WH + DRV. The results illustrate that Misaka’s and Shida’s models have good agreement with measured ones. Some deviations of flow stresses are observed when the strain exceeds, say, 0.3 for Shida’s equation. For the region of DRX, the drop of flow stress could be expressed as the following equation [13 equation [13–15] 15]:: ε > εp ,
σ DRX DRX = (σ p − σ ss ss )
ε < εp ,
σ DRX DRX = 0
XDRX − Xεp
1 − Xεp
(13) (14)
where σ ss is the steady state stress achieved at larger strains and Xεp the volume fraction of DRX at peak strain. X DRX DRX the volume fraction of DRX at any strain. The equation for volume fraction of dynamically recrystallized grain might be proposed as follows: m
XDRX = 1 − exp −
ε − εc ε∗
(15)
where m is the Avrami’is constant. εc represents the critical strain for initiating dynamic recrystallization and ε ∗ the flow stress strain for maximum softening rate. The change of fl is attributed to the evolution of microstructure such as dynamic recrystallization (DRX). Thus, the evolution of DRX fl ow curve, such as work can be analyzed from the slope of fl hardening rate. The onset and fi nish of DRX and the strain for maximum softening rate (ε∗ ) can be decided from the inflection points of work hardening r ate-strain curves or work hardening rate-stress curves [13,14] [13,14].. In the expression of , most researchers replace ε∗ with ε0.5 , which means X DRX DRX the strain for 50% recrystallization. Eq. recrystallization. Eq. (15), (15), which is modified based on the Avrami’s equation, means that X DRX DRX de∗ pends on the strain for maximum softening rate, ε as well as the coef ficient, m , critical strain, ε c and applied strain ε .
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DRX is a continuous process of deformation, nucleation of grains and subsequent migration of grain boundaries, X DRX DRX increases with the strain increased. As the strain increases, X DRX DRX reaches to a constant value, 100%. At X DRX = 100%, the fl ow stress reaches to steady state. It should be noted that most of previous works did not calculate the volume fraction of dynamically recrystallized grains [1–4,7] 4,7].. The flow stress in DRX region, σ DRX DRX , means a drop of stress due to the DRX. Thus, σ DRX DRX is proportional to the DRX volume fraction which is corrected with X εp . The total flow stress ( σ total total ) can be expressed with subtraction form.
o
1100 C, 0.5/sec
100
o
1000 C, 0.5/sec
) % ( 80 X R D f 60 o n o i t c a 40 r F e m u 20 l o V
o
900 C, 0.5/sec
0 0. 2
0 .4
0 . 6 0 .8 1
2
4
6
8
Strain Fig. 3. Calculated volume fractions of DRX obtained at various deformation conditions.
The validity of Eq. Eq. (15) can (15) can be confirmed through the application for other steels, such as austenitic stainless steels [15] steels [15].. Fig. 3 shows 3 shows the calculated volume fraction of DRX at different temperatures. The DRX is thermally activated so that the X DRX DRX increased with temperature. Also, Since the
(a)
σ total total = σ (WH+DRV) − σ DRX DRX
(16)
Fig. 4 shows Fig. shows the measur measured ed and predic predicted ted stress stress–strain curves for AISI 4140 steel at various deformation conditions. Fig. tions. Fig. 4(a) 4(a) illustrates the concept of work hardening (WH), dynamic recovery (DRV) and dynamic recrystallization (DRX) during hot deformation, previously explained. One can understand the procedure for calculating the flow stress by using Eqs. using Eqs. (12)–(16) (16).. Fig. 4(b) 4(b) illustrates that, in all cases, the predicted stress –strain curves by using the modified Voce’s equation are in good agreement with the measured ones.
180 o
1000 C, 5/sec
(a)
150
) a P M ( s s e r t S
200
o
900 C, 0.5s
Compression
120
-1
150
90
) a P M ( s 100 s e r t S
Calculated for WH+DRV region Calculated for DRX region Flow Stress
60 30
50
0 0
1
2
3
4
5
0 0.0
Strain
Torsion
Developed Eq. Shida's Eq. Misaka's Eq.
0.5
1.0
0.5
2.0
Strain (b)
180
o
Calculated
(b)
-1
1000 C, 5s o
150
o
o
1100 C, 5s o
) a P M ( s s e r t S
90 60
0 1
2
3
4
90 Torsion
60
30
Experimental
0
-1
120
-1
120
30
1000 C, 0.5s
Compression
-1
1100 C, 0.5s
) a P M ( s s e r t S
150
-1
1000 C, 0.5s
5
Strain Fig. 4. Comparison of measured and predicted constitutive relations for AISI 4140 steel deformed (a) at 1000 ◦ C and 5s−1 and (b) at various deformation conditions.
0 0.0
Developed Eq. Shida's Eq. Misaka's Eq.
0 .5
1.0
1 .5
2.0
Strain Fig. 5. Comparison of measured and predicted constitutive relations for AISI 4140 steel deformed (a) at 900 ◦ C and 0.5s−1 and (b) at 1000 ◦ C and 0.5 0.5 s−1 .
S.I. Kim et al. / Journal Journal of Materials Materials Processing Processing Technol Technology ogy 140 (2003) 84–89
Fig. 5 shows Fig. hows the measur measured ed and predic predicted ted flow stress stress curve curvess in large strain range when the three different types of constitutive equations are used for prediction. The flow stress curves calculated by using Misaka ’s equation agree with the measured ones with some extent of error in comparison to the flow stress curve obtained from Shida’s equation. It seems Misaka ’s equation does not re flect recrystallization behavior properly. properly. The stress–strain curves predicted by using the modi fied Voce’s equation are in good agreement with experimentally measured ones. Thus, it can be deduced that the approach to obtain a constitutive equation applicable to large strain ranges ranges was fruitful and this modified equation might have a potential to be used for hot strip rolling process where more precise calculation of stress decrement due to dynamic recrystallization is important. It should be noted that the fl ow stress curves predicted by Shida’s equation yields rapid decrease as strain passes, say, 0.7, 0.7, which which means means the steel steel exper experien iences ces recrys recrystal talliz lizati ation. on. This This is the the reas reason on why why Shid Shidaa’s equati equation on set applic applicati ation on limita limitatio tions ns in strain. 5. Conclusions Conclusions
We have have deve develop loped ed the equati equations ons for predic predictin ting g flow stress stress curve for AISI 4140 (medium carbon and low alloy steel) using hot torsion and compression tests in the temperature rang rangee of 900 900–1100 ◦ C and and the the stra strain in rate rate rang rangee of 0.05 0.05–5 s−1 . The conclusions are summarized as follows: (1) It has been found out that the approach to obtain the develope developed d constituti constitutive ve equation equation applicable applicable to large large strain strain ranges at elevated temperatures was successful. (2) As far as strain rate is not concerned, it is better better to obtain required data for constitutive relation through torsion test, which allows a large strain and consequently simulation of recrystallization. (3) It is expected that, with further tests to examine the validity of the modified constitutive equation for high strain rate ranges, the proposed constitutive equation may be successfully applied to the accurate analysis of deformation (roll force and torque) of materials in hot strip and/or plate rolling process.
89
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