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Constitutive equations to predict high temperature flow stress in a Ti-modified austenitic stainless steel
Materials Science and Engineering A 500 (2009) 114–121
Contents lists available at ScienceDirect
Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea
Constitutive equations to predict high temperature flow stress in a Ti-modified austenitic stainless steel Sumantra Mandal ∗ , V. Rakesh, P.V. Sivaprasad, S. Venugopal, K.V. Kasiviswanathan Metallurgy and Materials Group, Indira Gandhi Centre for Atomic Research, Kalpakkam, Tamil Nadu 603102, India
a r t i c l e
i n f o
Article history: Received 20 August 2008 Received in revised form 4 September 2008 Accepted 4 September 2008 Keywords: Austenitic stainless steel Constitutive equation Flow stress Compensation of temperature and strain rate
a b s t r a c t The experimental stress–strain data from isothermal hot compression tests, in a wide range of temperatures (1123–1523 K) and strain rates (10−3 –102 s−1 ), were employed to develop constitutive equations in a Ti-modified austenitic stainless steel. The effects of temperature and strain rate on deformation behaviors were represented by Zener-Holloman parameter in an exponent type equation. The influence of strain was incorporated in the constitutive analysis by considering the effect of strain on material constants. The constitutive equation (considering the compensation of strain) could precisely predict the flow stress only at 0.1 and 1 s−1 strain rates. A modified constitutive equation (incorporating both the strain and strain rate compensation), on the other hand, could predict the flow stress throughout the entire temperatures and strain rates range except at 1123 K in 10 and 100 s−1 . The breakdown of the constitutive equation at these processing conditions is possibly due to adiabatic temperature rise during high strain rate deformation. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Austenitic stainless steels, primarily AISI 316 and its modifications, have been selected world-wide as prime candidate materials for fuel cladding and sub-assembly wrapper tubes of fast breeder reactors owing to their excellent elevated temperature mechanical properties, compatibility with liquid sodium and adequate resistance to void swelling [1]. For the 500 MWe fast breeder reactor project (PFBR) in India, a 15Cr–15Ni–2.2Mo–Ti modified austenitic stainless steel (commonly known as alloy D9) has been developed indigenously [2]. This is a candidate material for in-core applications as fuel cladding tube and hexagonal subassembly wrapper. Alloy D9 has to be processed through various thermo-mechanical treatments before it is fabricated into final component. Determination of the load to carry out these operations is of paramount importance. Load depends on flow stress of the materials besides the geometry of the die and friction at the tool–work piece interface. Therefore, prediction of hot deformation behavior linking process variables such as strain, strain rate and temperature to the flow stress of the deforming materials is necessary [3]. Constitutive equation is often used to represent flow behaviors of the metals and alloys in a form that is suitable to feed in computer code to model the materials response under the specified loading conditions [4]. In recent past, various analytical, phenomenological
∗ Corresponding author. Tel.: +91 44 27480118; fax: +91 44 27480075. E-mail address: [email protected] (S. Mandal). 0921-5093/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2008.09.019
and empirical models have been constructed to predict constitutive behavior in a wide range of metals and alloys. A phenomenological approach was proposed by Jonas et al. [5] where the flow stress is expressed by the hyperbolic laws in an Arrhenius type of equation. A strain-dependent parameter into the sine hyperbolic constitutive equation was introduced, by Sloof et al. [6], to predict the flow stress in a wrought magnesium alloy. A revised sine hyperbolic constitutive equation was adopted by incorporation of strain dependent term to predict elevated temperature flow behavior in 42CrMo steel and boron micro alloying TiAl–Cr–V alloys [4,7]. A comprehensive review on constitutive analysis in hot working of austenitic stainless steel, carbon and alloy steels, ferritic steel and Al alloys could be found in [8]. The objective of this study is to establish the relationship between the flow stress, strain, strain rate and temperature to predict high temperature flow behavior of alloy D9. Toward this end, isothermal hot compression tests were conducted in a wide range of strain rates and temperatures. The experimental stress–strain data were then employed to derive constitutive equation relating flow stress, strain rate and temperature considering compensation of strain and strain rate. Finally, validity of the developed constitutive equation was examined for the entire experimental range. 2. Experimental The chemical composition of alloy D9 used in the present investigation is given in Table 1. Cylindrical specimens of 10 mm diameter and 15 mm height were machined to carry out
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Table 1 Chemical composition (in wt.%) of 15Cr–15Ni–2.2Mo–Ti modified austenitic stainless steel (alloy D9). C Mn Si S P Cr Ni Mo Ti B Co N
0.052 1.509 0.505 0.002 0.011 15.051 15.068 2.248 0.31 0.001 0.01 0.006
compression testing. The isothermal hot compression tests were conducted using a computer-controlled servo hydraulic testing machine (DARTEC, Stourbridge, UK) with a maximum load capacity of 100 kN. The machine is equipped with a control system to impose exponential decay of the actuator speed to obtain constant true strain rates. A resistance heating split furnace with SiC heating elements was used to surround the platens and specimen. The specimens were coated with a borosilicate glass paste that acted as a lubricant as well as a protective coating. The testing temperatures ranged from 1123 to 1523 K at an interval of 50 K and at constant true strain rates of 0.001, 0.01, 0.1, 1, 10 and 100 s−1 . The specimens were deformed and subsequently quenched in water. The load-stroke data obtained in compression tests were processed to obtain true stress-true plastic strain using standard equations. 3. Results and discussion The experimental flow curves of alloy D9 at two different strain rates are shown in Fig. 1. It could be observed that the influence of temperature and strain rate on flow stress is significant. The flow stress increases with decrease in temperature and increases in strain rate. The flow curve typically shows work hardening region followed by dynamic softening due to recovery/recrystallization. The work hardening is predominant at lower temperatures and higher strain rates. On the contrary, the extent of dynamic softening is more at higher temperatures and lower strain rates. This is due to the fact that higher temperatures and lower strain rates offer higher mobility to the grain boundary and longer time for nucleation and growth of dynamically recrystallized grains. 3.1. Constitutive equations for flow stress prediction The correlation between the flow stress (), temperature (T) and ˙ particularly at high temperatures, could be expressed strain rate (ε), by an Arrhenius type equation [9]. Further, the effects of temperature and strain rate on deformation behavior could be represented by Zener-Holloman parameter (Z) in an exponent type equation [10]. These are mathematically expressed as Z = ε˙ exp
Q
(1)
RT
Q
ε˙ = AF() exp − where F() ==
(2)
RT
n exp(ˇ) [sinh(˛)]n
˛ < 0.8 ˛ > 1.2 for all
Fig. 1. Flow curves of alloy D9 at various temperatures with strain rate of (a) 0.001 s−1 and (b) 10 s−1 .
Here, R is the universal gas constant (8.31 J mole−1 K−1 ); T is the absolute temperature in K; Q is the activation energy (kJ mol−1 ); A, ˛ and n are the materials constants, ˛ = ˇ/n. 3.2. Determination of materials constants Flow stress vs true strain data from the compression tests at various processing conditions was used to evaluate the materials constants of the constitutive equations. The following are the evaluation procedure of material constants at a true strain of 0.1. For low and high stress levels, substituting the value of F() in Eq. (2) gives the following relationships respectively, ε˙ = B n
(3)
ε˙ = C exp(ˇ)
(4)
where, B and C are the material constants. Logarithm of both sides of Eqs. (3) and (4) yields, ln() =
1 1 ˙ − ln(B) ln(ε) n n
(5)
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S. Mandal et al. / Materials Science and Engineering A 500 (2009) 114–121 Table 2 Coefficients of the polynomial for ˛, ˇ, n, Q and ln A. ˛
ˇ
n
Q
ln A
C0 = 0.013 C1 = −0.066 C2 = 0.306 C3 = −0.617 C4 = 0.454
D0 = 0.128 D1 = −0.816 D2 = 3.639 D3 = −7.389 D4 = 5.539
E0 = 10.351 E1 = −20.0352 E2 = 66.445 E3 = −127.563 E4 = 101.431
F0 = 697.946 F1 = −881.611 F2 = 3665.192 F3 = −7625.935 F4 = 6145.841
G0 = 56.933 G1 = −42.379 G2 = 139.437 G3 = −266.392 G4 = 226.276
The value of Q can be derived from the slopes in a plot of ln[sinh(˛)] vs. 1/T (Fig. 3a). The value of the Q was determined by averaging the values of Q under different strain rates. At 0.1 true strain, the value of Q was found to be 638.57 kJ/mol. The value of A at a particular strain could be obtained by plotting ˙ As shown in Fig. 3b, the relationship between ln[sinh(˛)] and ln ε. the value of A at 0.1 strain was found to be 2.32 × 1023 . 3.3. Compensation of strain It is assumed that influence of strain on high temperature flow behavior is insignificant and thereby would not be considered in Eq. (1). However, as discussed, effect of strain is significant in the lower hot working temperature regime (Fig. 1). Hence, compensation of
Fig. 2. Evaluating the value of (a) n by plotting ln vs ln ε˙ and (b) ˇ by plotting ˙ vs ln ε.
=
1 1 ˙ − ln(C) ln(ε) ˇ ˇ
(6)
The value of n and ˇ can be obtained from the slope of the lines in the ln − ln ε˙ plot and − ln ε˙ plot, respectively (Fig. 2). It is apparent that the lines are almost parallel leading us to observe that the slope of the lines consequently varies in a very small range. The slight variation in the slope of the lines could be attributed to scattering in the experimental data points. The mean value of the slopes was taken as the value of n and ˇ which was found to be 8.8415 and 0.0745 MPa−1 , respectively. This gives the value of ˛ = ˇ/n = 0.0084 MPa−1 . For low as well as high stress levels, Eq. (2) can be written as
Q
ε˙ = A[sinh(˛)]n exp −
RT
(7)
Taking the logarithm of both sides of the above Eq. (7) gives ln[sinh(˛)] =
ln ε˙ Q ln A + − n n (nRT )
(8)
For a particular strain rate, differentiating Eq. (8) gives Q = Rn
d{ln[sinh ˛]} d(1/T )
(9)
Fig. 3. Evaluating the value of (a) Q by plotting ln vs ln ε˙ and (b) ˇ by plotting ˙ vs ln ε.
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117
Fig. 4. Variation of (a) ˛, (b) ˇ, (c) ln A, (d) n and (e) Q with true strain.
strain should be taken into account in order to derive constitutive equations to accurately predict the flow stress. The influence of strain in the constitutive equation is incorporated by assuming that the material constants (i.e. ˛, ˇ, n, Q and A) are polynomial function of strains. The values of the material constant were evaluated at various strains (in the range 0.05–0.5) at an interval of 0.05. These values were then employed to fit the polynomial. The order of the polynomial was varied from 1 to 9. A fourth order polynomial, as shown in Eq. (10), was found to represent the influence of strain on material constants with a very good correlation and generalization (Fig. 4). Higher order (i.e. >4) polynomial was found to over-fit
thus losing its ability of true representation and generalization. The coefficients of the polynomial are given in Table 2. ˛ ˇ n Q ln A
= C0 + C1 ε + C2 ε2 + C3 ε3 + C4 ε4 = D0 + D1 ε + D2 ε2 + D3 ε3 + D4 ε4 = E0 + E1 ε + E2 ε2 + E3 ε3 + E4 ε4 = F0 + F1 ε + F2 ε2 + F3 ε3 + F4 ε4 = G0 + G1 ε + G2 ε2 + G3 ε3 + G4 ε4
(10)
Once the materials constants are evaluated, the flow stress at a particular strain can be predicted. The constitutive equation that relates flow stress and Zener-Holloman parameter can be written
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S. Mandal et al. / Materials Science and Engineering A 500 (2009) 114–121
Fig. 5. Comparison between the experimental and predicted flow stress from the constitutive equation (considering the compensation of strain) at strain rate (a) 0.001 s−1 , (b) 0.01 s−1 , (c) 0.1 s−1 , (d) 1 s−1 , (e) 10 s−1 and (f) 100 s−1 .
in following form (considering the Eq. (1) and Eq. (7)): =
1 ln ˛
Z 1/n A
+
2/n Z A
1/2
+1
(11)
3.4. Verification of constitutive equation The developed constitutive equation (considering the compensation of strain) was verified by comparing the experimental and
predicted data (Fig. 5). It could be observed that prediction is good at strain rates of 0.1 and 1 s−1 (Fig. 5c and d). However, a significant deviation in prediction is observed below 0.1 s−1 (Fig. 5a and b) and above 1 s−1 (Fig. 5e and f). A close look at Fig. 5a and b reveals that the developed constitutive equation consistently over predicted the flow stress value below 0.1 s−1 . On the other hand, an obvious underestimation in flow stress value is observed above 1 s−1 (Fig. 5e and f) at all temperatures except 1123 K. These trends lead us to believe that a modification of the developed constitu-
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119
Fig. 6. Comparison between the experimental and predicted flow stress from the constitutive equation (considering the compensation of strain and strain rate) at strain rate (a) 0.001 s−1 , (b) 0.01 s−1 , (c) 0.1 s−1 , (d) 1 s−1 , (e) 10 s−1 and (f) 100 s−1 .
tive equation, in terms of compensation of strain rate is required to improve its predictability through the all strain rates. 3.5. Modification of constitutive equation The modification of the constitutive equation was carried out by compensation of strain rate in Zener-Holloman parameter (Z). This
was done by modifying the exponent of strain rate in Z. A similar approach was adopted by Lin et al. to compensate the strain rate in 42CrMo steel [11]. It should be noted here that exponent of strain rate in original Zener-Holloman parameter (Z) is 1 (Eq. (1)). Three different values of exponent (i.e. 4/3, 3/2 and 5/3) of strain rate were tried in order to find the most suitable form of modified Z. An exponent of 4/3 was found to be optimum to compensate the strain rate for the investigated alloy. The modified Zener-Holloman parameter
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S. Mandal et al. / Materials Science and Engineering A 500 (2009) 114–121
(Z*) and the modified constitutive equation (with compensation of both the strain and strain rate) could be expressed as follows: Z ∗ = ε˙ 4/3 exp 1 = ln ˛
Q
(12)
RT
Z ∗ 1/n A
+
2/n Z∗ A
1/2 +1
(13)
where Q, ˛ and A could be obtained from Eq. (10). 3.6. Verification of the modified constitutive equation The predictability of the modified constitutive equation was assessed by comparing the experimental and predicted data as shown in Fig. 6. It could be observed that predicted flow stress value from the modified constitutive equation could track the experimental data throughout the entire strain rate range. Only in two processing conditions (i.e. at 1123 K in 10 and 100 s−1 ), a notable variation between experimental and predicted flow stress data could be observed (Fig. 6e and f). It should be noted that these are the only two processing conditions at high strain rates (≥10 s−1 ) where prediction from the constitutive equation (Eq. (11)) shows the reverse trend (i.e. an overestimation in flow stress prediction, see Fig. 5e and f). The reason is not quite clear to us. Possibly, temperature rise due to deformation heating at high strain rates (i.e. 10 and 100 s−1 ) leads to a significant part of flow softening since the test time is too short to allow the heat to transfer [12]. It has been suggested that the stress–strain data obtained from hot working tests above 10 s−1 are not suitable for constitutive analysis unless adiabatic temperature changes are accounted [6]. It may be argued here that deformation heating and subsequent flow stress reduction should take place at other temperatures also at these high strain rates. Accordingly, the modified constitutive equation should fail at other temperatures as well. However, the modified equation could predict the flow stress accurately and precisely at other temperatures at these high strain rates (Fig. 6e and f). Possibly, the influence of deformation heating on flow softening in these high temperature regimes is not so significant since available thermal activation energy is already high to induce flow softening. Further work needs to be done in this direction to draw a firm conclusion. The predictability of the constitutive equation is also quantified employing standard statistical parameters such as correlation coefficient (R) and average absolute relative error (AARE). These are expressed as
N R=
N
i=1
¯ i − P) ¯ (Ei − E)(P
¯ (E − E) i=1 i
N
¯ (P − P) i=1 i
(14) 2
1 Ei − Pi E × 100 N i N
AARE(%) =
2
(15)
i=1
where E is the experimental finding and P is the predicted value obtained from the constitutive equation. E¯ and P¯ are the mean values of E and P, respectively. N is the number of data employed in the investigation. The correlation coefficient is a commonly used statistical parameter and provides information about the strength of linear relationship between the observed and the calculated values. Sometimes higher value of R may not necessarily indicate a better performance [13] because of the tendency of the model/equation to be biased towards higher or lower values. The AARE is computed through a term-by-term comparison of the relative error and therefore is an unbiased statistical parameter for measuring the predictability of a model/equation [14].
Fig. 7. Correlation between the experimental and predicted flow stress data from the modified constitutive equation.
As can be seen from Fig. 7, a good correlation between experimental and predicted data is obtained. Comparatively larger scattering in the data points above 400 MPa can be seen. This is due to break down of the modified constitutive equation at low temperature (1123 K) and high strain rates (≥10 s−1 ) as discussed in the previous section. In spite of this scattering, the AARE was found to be 6.74%, which reflects the good prediction capabilities of the proposed constitutive equation. 4. Conclusions Constitutive analysis of alloy D9 was carried out by performing hot compression tests in a wide range of temperatures (1123–1523 K) and strain rates (0.001–100 s−1 ). The following are the conclusions: • The influence of strain in the constitutive analysis was incorporated by considering the effect of strain on material constants (i.e. ˛, ˇ, n, Q and A). A fourth order polynomial was found suitable to represent the influence of strain on material constants with very good correlation and generalization. • The constitutive equation (considering only the compensation of strain) was found to predict flow stress precisely at strain rates of 0.1 and 1 s−1 . However, a significant deviation in the prediction is observed below 0.1 s−1 and above 1 s−1 . • The revised constitutive equation, incorporating the strain and strain rate compensation, could predict the flow stress throughout the entire temperatures and strain rates range (except at 1123 K in 10 and 100 s−1 ). The breakdown of the constitutive equation at these processing conditions is believed to be due to adiabatic temperature rise during this high strain rate deformation. Acknowledgements The authors gratefully acknowledge Dr. Baldev Raj, Director, Indira Gandhi Centre for Atomic Research, for his constant encouragement during the investigation. References [1] S.L. Mannan, P.V. Sivaprasad, in: K.H. Jurgen Buschow, R.W. Cahn, M.C. Flemings, B. Ilschner, E.J. Kramer, S. Mahajan (Eds.), Encyclopedia of Materials Science and Technology, vol. 3, Elsevier, New York, 2001, pp. 2857–2865.
S. Mandal et al. / Materials Science and Engineering A 500 (2009) 114–121 [2] P.V. Sivaprasad, Sumantra Mandal, S. Venugopal, C. Narayanan, V. Shanmugam, Baldev Raj, Transaction of the Indian Institute of Metals 59 (2006) 437–445. [3] Sumantra Mandal, P.V. Sivaprasad, S. Venugopal, K.P.N. Murthy, Baldev Raj, Materials Science and Engineering A 485 (2008) 571–580. [4] Y.-C. Lin, M.-S. Chen, J. Zhang, Materials Science and Engineering A 499 (2009) 88–92. [5] J. Jonas, C.M. Sellars, W.J.McG. Tegart, International Metallurgy Review 14 (1969) 1. [6] F.A. Slooff, J. Zhou, J. Duszczyk, L. Katgerman, Scripta Materialia 57 (2007) 759–762. [7] Z.J. Pu, K.H. Wu, J. Shai, D. Zou, Materials Science and Engineering A 192–193 (1995) 780–787.
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[8] H.J. McQueen, N.D. Ryan, Materials Science and Engineering A 322 (2002) 43–63. [9] C.M. Sellars, W.J. McTegart, Acta Metallurgica 14 (1966) 1136–1138. [10] C. Zener, H. Hollomon, J. Appl. Phys. 15 (1944) 22. [11] Y.-C. Lin, M.-S. Chen, J. Zhang, Computational Materials Science 42 (2008) 470–477. [12] Y.C. Lin, M.-S. Chen, J. Zhong, Mechanics Research Communications 35 (2008) 142–150. [13] M.P. Phaniraj, A.K. Lahiri, Journal of Material Processes and Technology 141 (2003) 219–227. [14] S. Srinivasulu, A. Jain, Applied Soft Computing 6 (2006) 295–306.
Contents lists available at ScienceDirect
Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea
Constitutive equations to predict high temperature flow stress in a Ti-modified austenitic stainless steel Sumantra Mandal ∗ , V. Rakesh, P.V. Sivaprasad, S. Venugopal, K.V. Kasiviswanathan Metallurgy and Materials Group, Indira Gandhi Centre for Atomic Research, Kalpakkam, Tamil Nadu 603102, India
a r t i c l e
i n f o
Article history: Received 20 August 2008 Received in revised form 4 September 2008 Accepted 4 September 2008 Keywords: Austenitic stainless steel Constitutive equation Flow stress Compensation of temperature and strain rate
a b s t r a c t The experimental stress–strain data from isothermal hot compression tests, in a wide range of temperatures (1123–1523 K) and strain rates (10−3 –102 s−1 ), were employed to develop constitutive equations in a Ti-modified austenitic stainless steel. The effects of temperature and strain rate on deformation behaviors were represented by Zener-Holloman parameter in an exponent type equation. The influence of strain was incorporated in the constitutive analysis by considering the effect of strain on material constants. The constitutive equation (considering the compensation of strain) could precisely predict the flow stress only at 0.1 and 1 s−1 strain rates. A modified constitutive equation (incorporating both the strain and strain rate compensation), on the other hand, could predict the flow stress throughout the entire temperatures and strain rates range except at 1123 K in 10 and 100 s−1 . The breakdown of the constitutive equation at these processing conditions is possibly due to adiabatic temperature rise during high strain rate deformation. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Austenitic stainless steels, primarily AISI 316 and its modifications, have been selected world-wide as prime candidate materials for fuel cladding and sub-assembly wrapper tubes of fast breeder reactors owing to their excellent elevated temperature mechanical properties, compatibility with liquid sodium and adequate resistance to void swelling [1]. For the 500 MWe fast breeder reactor project (PFBR) in India, a 15Cr–15Ni–2.2Mo–Ti modified austenitic stainless steel (commonly known as alloy D9) has been developed indigenously [2]. This is a candidate material for in-core applications as fuel cladding tube and hexagonal subassembly wrapper. Alloy D9 has to be processed through various thermo-mechanical treatments before it is fabricated into final component. Determination of the load to carry out these operations is of paramount importance. Load depends on flow stress of the materials besides the geometry of the die and friction at the tool–work piece interface. Therefore, prediction of hot deformation behavior linking process variables such as strain, strain rate and temperature to the flow stress of the deforming materials is necessary [3]. Constitutive equation is often used to represent flow behaviors of the metals and alloys in a form that is suitable to feed in computer code to model the materials response under the specified loading conditions [4]. In recent past, various analytical, phenomenological
∗ Corresponding author. Tel.: +91 44 27480118; fax: +91 44 27480075. E-mail address: [email protected] (S. Mandal). 0921-5093/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2008.09.019
and empirical models have been constructed to predict constitutive behavior in a wide range of metals and alloys. A phenomenological approach was proposed by Jonas et al. [5] where the flow stress is expressed by the hyperbolic laws in an Arrhenius type of equation. A strain-dependent parameter into the sine hyperbolic constitutive equation was introduced, by Sloof et al. [6], to predict the flow stress in a wrought magnesium alloy. A revised sine hyperbolic constitutive equation was adopted by incorporation of strain dependent term to predict elevated temperature flow behavior in 42CrMo steel and boron micro alloying TiAl–Cr–V alloys [4,7]. A comprehensive review on constitutive analysis in hot working of austenitic stainless steel, carbon and alloy steels, ferritic steel and Al alloys could be found in [8]. The objective of this study is to establish the relationship between the flow stress, strain, strain rate and temperature to predict high temperature flow behavior of alloy D9. Toward this end, isothermal hot compression tests were conducted in a wide range of strain rates and temperatures. The experimental stress–strain data were then employed to derive constitutive equation relating flow stress, strain rate and temperature considering compensation of strain and strain rate. Finally, validity of the developed constitutive equation was examined for the entire experimental range. 2. Experimental The chemical composition of alloy D9 used in the present investigation is given in Table 1. Cylindrical specimens of 10 mm diameter and 15 mm height were machined to carry out
S. Mandal et al. / Materials Science and Engineering A 500 (2009) 114–121
115
Table 1 Chemical composition (in wt.%) of 15Cr–15Ni–2.2Mo–Ti modified austenitic stainless steel (alloy D9). C Mn Si S P Cr Ni Mo Ti B Co N
0.052 1.509 0.505 0.002 0.011 15.051 15.068 2.248 0.31 0.001 0.01 0.006
compression testing. The isothermal hot compression tests were conducted using a computer-controlled servo hydraulic testing machine (DARTEC, Stourbridge, UK) with a maximum load capacity of 100 kN. The machine is equipped with a control system to impose exponential decay of the actuator speed to obtain constant true strain rates. A resistance heating split furnace with SiC heating elements was used to surround the platens and specimen. The specimens were coated with a borosilicate glass paste that acted as a lubricant as well as a protective coating. The testing temperatures ranged from 1123 to 1523 K at an interval of 50 K and at constant true strain rates of 0.001, 0.01, 0.1, 1, 10 and 100 s−1 . The specimens were deformed and subsequently quenched in water. The load-stroke data obtained in compression tests were processed to obtain true stress-true plastic strain using standard equations. 3. Results and discussion The experimental flow curves of alloy D9 at two different strain rates are shown in Fig. 1. It could be observed that the influence of temperature and strain rate on flow stress is significant. The flow stress increases with decrease in temperature and increases in strain rate. The flow curve typically shows work hardening region followed by dynamic softening due to recovery/recrystallization. The work hardening is predominant at lower temperatures and higher strain rates. On the contrary, the extent of dynamic softening is more at higher temperatures and lower strain rates. This is due to the fact that higher temperatures and lower strain rates offer higher mobility to the grain boundary and longer time for nucleation and growth of dynamically recrystallized grains. 3.1. Constitutive equations for flow stress prediction The correlation between the flow stress (), temperature (T) and ˙ particularly at high temperatures, could be expressed strain rate (ε), by an Arrhenius type equation [9]. Further, the effects of temperature and strain rate on deformation behavior could be represented by Zener-Holloman parameter (Z) in an exponent type equation [10]. These are mathematically expressed as Z = ε˙ exp
Q
(1)
RT
Q
ε˙ = AF() exp − where F() ==
(2)
RT
n exp(ˇ) [sinh(˛)]n
˛ < 0.8 ˛ > 1.2 for all
Fig. 1. Flow curves of alloy D9 at various temperatures with strain rate of (a) 0.001 s−1 and (b) 10 s−1 .
Here, R is the universal gas constant (8.31 J mole−1 K−1 ); T is the absolute temperature in K; Q is the activation energy (kJ mol−1 ); A, ˛ and n are the materials constants, ˛ = ˇ/n. 3.2. Determination of materials constants Flow stress vs true strain data from the compression tests at various processing conditions was used to evaluate the materials constants of the constitutive equations. The following are the evaluation procedure of material constants at a true strain of 0.1. For low and high stress levels, substituting the value of F() in Eq. (2) gives the following relationships respectively, ε˙ = B n
(3)
ε˙ = C exp(ˇ)
(4)
where, B and C are the material constants. Logarithm of both sides of Eqs. (3) and (4) yields, ln() =
1 1 ˙ − ln(B) ln(ε) n n
(5)
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S. Mandal et al. / Materials Science and Engineering A 500 (2009) 114–121 Table 2 Coefficients of the polynomial for ˛, ˇ, n, Q and ln A. ˛
ˇ
n
Q
ln A
C0 = 0.013 C1 = −0.066 C2 = 0.306 C3 = −0.617 C4 = 0.454
D0 = 0.128 D1 = −0.816 D2 = 3.639 D3 = −7.389 D4 = 5.539
E0 = 10.351 E1 = −20.0352 E2 = 66.445 E3 = −127.563 E4 = 101.431
F0 = 697.946 F1 = −881.611 F2 = 3665.192 F3 = −7625.935 F4 = 6145.841
G0 = 56.933 G1 = −42.379 G2 = 139.437 G3 = −266.392 G4 = 226.276
The value of Q can be derived from the slopes in a plot of ln[sinh(˛)] vs. 1/T (Fig. 3a). The value of the Q was determined by averaging the values of Q under different strain rates. At 0.1 true strain, the value of Q was found to be 638.57 kJ/mol. The value of A at a particular strain could be obtained by plotting ˙ As shown in Fig. 3b, the relationship between ln[sinh(˛)] and ln ε. the value of A at 0.1 strain was found to be 2.32 × 1023 . 3.3. Compensation of strain It is assumed that influence of strain on high temperature flow behavior is insignificant and thereby would not be considered in Eq. (1). However, as discussed, effect of strain is significant in the lower hot working temperature regime (Fig. 1). Hence, compensation of
Fig. 2. Evaluating the value of (a) n by plotting ln vs ln ε˙ and (b) ˇ by plotting ˙ vs ln ε.
=
1 1 ˙ − ln(C) ln(ε) ˇ ˇ
(6)
The value of n and ˇ can be obtained from the slope of the lines in the ln − ln ε˙ plot and − ln ε˙ plot, respectively (Fig. 2). It is apparent that the lines are almost parallel leading us to observe that the slope of the lines consequently varies in a very small range. The slight variation in the slope of the lines could be attributed to scattering in the experimental data points. The mean value of the slopes was taken as the value of n and ˇ which was found to be 8.8415 and 0.0745 MPa−1 , respectively. This gives the value of ˛ = ˇ/n = 0.0084 MPa−1 . For low as well as high stress levels, Eq. (2) can be written as
Q
ε˙ = A[sinh(˛)]n exp −
RT
(7)
Taking the logarithm of both sides of the above Eq. (7) gives ln[sinh(˛)] =
ln ε˙ Q ln A + − n n (nRT )
(8)
For a particular strain rate, differentiating Eq. (8) gives Q = Rn
d{ln[sinh ˛]} d(1/T )
(9)
Fig. 3. Evaluating the value of (a) Q by plotting ln vs ln ε˙ and (b) ˇ by plotting ˙ vs ln ε.
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Fig. 4. Variation of (a) ˛, (b) ˇ, (c) ln A, (d) n and (e) Q with true strain.
strain should be taken into account in order to derive constitutive equations to accurately predict the flow stress. The influence of strain in the constitutive equation is incorporated by assuming that the material constants (i.e. ˛, ˇ, n, Q and A) are polynomial function of strains. The values of the material constant were evaluated at various strains (in the range 0.05–0.5) at an interval of 0.05. These values were then employed to fit the polynomial. The order of the polynomial was varied from 1 to 9. A fourth order polynomial, as shown in Eq. (10), was found to represent the influence of strain on material constants with a very good correlation and generalization (Fig. 4). Higher order (i.e. >4) polynomial was found to over-fit
thus losing its ability of true representation and generalization. The coefficients of the polynomial are given in Table 2. ˛ ˇ n Q ln A
= C0 + C1 ε + C2 ε2 + C3 ε3 + C4 ε4 = D0 + D1 ε + D2 ε2 + D3 ε3 + D4 ε4 = E0 + E1 ε + E2 ε2 + E3 ε3 + E4 ε4 = F0 + F1 ε + F2 ε2 + F3 ε3 + F4 ε4 = G0 + G1 ε + G2 ε2 + G3 ε3 + G4 ε4
(10)
Once the materials constants are evaluated, the flow stress at a particular strain can be predicted. The constitutive equation that relates flow stress and Zener-Holloman parameter can be written
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Fig. 5. Comparison between the experimental and predicted flow stress from the constitutive equation (considering the compensation of strain) at strain rate (a) 0.001 s−1 , (b) 0.01 s−1 , (c) 0.1 s−1 , (d) 1 s−1 , (e) 10 s−1 and (f) 100 s−1 .
in following form (considering the Eq. (1) and Eq. (7)): =
1 ln ˛
Z 1/n A
+
2/n Z A
1/2
+1
(11)
3.4. Verification of constitutive equation The developed constitutive equation (considering the compensation of strain) was verified by comparing the experimental and
predicted data (Fig. 5). It could be observed that prediction is good at strain rates of 0.1 and 1 s−1 (Fig. 5c and d). However, a significant deviation in prediction is observed below 0.1 s−1 (Fig. 5a and b) and above 1 s−1 (Fig. 5e and f). A close look at Fig. 5a and b reveals that the developed constitutive equation consistently over predicted the flow stress value below 0.1 s−1 . On the other hand, an obvious underestimation in flow stress value is observed above 1 s−1 (Fig. 5e and f) at all temperatures except 1123 K. These trends lead us to believe that a modification of the developed constitu-
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Fig. 6. Comparison between the experimental and predicted flow stress from the constitutive equation (considering the compensation of strain and strain rate) at strain rate (a) 0.001 s−1 , (b) 0.01 s−1 , (c) 0.1 s−1 , (d) 1 s−1 , (e) 10 s−1 and (f) 100 s−1 .
tive equation, in terms of compensation of strain rate is required to improve its predictability through the all strain rates. 3.5. Modification of constitutive equation The modification of the constitutive equation was carried out by compensation of strain rate in Zener-Holloman parameter (Z). This
was done by modifying the exponent of strain rate in Z. A similar approach was adopted by Lin et al. to compensate the strain rate in 42CrMo steel [11]. It should be noted here that exponent of strain rate in original Zener-Holloman parameter (Z) is 1 (Eq. (1)). Three different values of exponent (i.e. 4/3, 3/2 and 5/3) of strain rate were tried in order to find the most suitable form of modified Z. An exponent of 4/3 was found to be optimum to compensate the strain rate for the investigated alloy. The modified Zener-Holloman parameter
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(Z*) and the modified constitutive equation (with compensation of both the strain and strain rate) could be expressed as follows: Z ∗ = ε˙ 4/3 exp 1 = ln ˛
Q
(12)
RT
Z ∗ 1/n A
+
2/n Z∗ A
1/2 +1
(13)
where Q, ˛ and A could be obtained from Eq. (10). 3.6. Verification of the modified constitutive equation The predictability of the modified constitutive equation was assessed by comparing the experimental and predicted data as shown in Fig. 6. It could be observed that predicted flow stress value from the modified constitutive equation could track the experimental data throughout the entire strain rate range. Only in two processing conditions (i.e. at 1123 K in 10 and 100 s−1 ), a notable variation between experimental and predicted flow stress data could be observed (Fig. 6e and f). It should be noted that these are the only two processing conditions at high strain rates (≥10 s−1 ) where prediction from the constitutive equation (Eq. (11)) shows the reverse trend (i.e. an overestimation in flow stress prediction, see Fig. 5e and f). The reason is not quite clear to us. Possibly, temperature rise due to deformation heating at high strain rates (i.e. 10 and 100 s−1 ) leads to a significant part of flow softening since the test time is too short to allow the heat to transfer [12]. It has been suggested that the stress–strain data obtained from hot working tests above 10 s−1 are not suitable for constitutive analysis unless adiabatic temperature changes are accounted [6]. It may be argued here that deformation heating and subsequent flow stress reduction should take place at other temperatures also at these high strain rates. Accordingly, the modified constitutive equation should fail at other temperatures as well. However, the modified equation could predict the flow stress accurately and precisely at other temperatures at these high strain rates (Fig. 6e and f). Possibly, the influence of deformation heating on flow softening in these high temperature regimes is not so significant since available thermal activation energy is already high to induce flow softening. Further work needs to be done in this direction to draw a firm conclusion. The predictability of the constitutive equation is also quantified employing standard statistical parameters such as correlation coefficient (R) and average absolute relative error (AARE). These are expressed as
N R=
N
i=1
¯ i − P) ¯ (Ei − E)(P
¯ (E − E) i=1 i
N
¯ (P − P) i=1 i
(14) 2
1 Ei − Pi E × 100 N i N
AARE(%) =
2
(15)
i=1
where E is the experimental finding and P is the predicted value obtained from the constitutive equation. E¯ and P¯ are the mean values of E and P, respectively. N is the number of data employed in the investigation. The correlation coefficient is a commonly used statistical parameter and provides information about the strength of linear relationship between the observed and the calculated values. Sometimes higher value of R may not necessarily indicate a better performance [13] because of the tendency of the model/equation to be biased towards higher or lower values. The AARE is computed through a term-by-term comparison of the relative error and therefore is an unbiased statistical parameter for measuring the predictability of a model/equation [14].
Fig. 7. Correlation between the experimental and predicted flow stress data from the modified constitutive equation.
As can be seen from Fig. 7, a good correlation between experimental and predicted data is obtained. Comparatively larger scattering in the data points above 400 MPa can be seen. This is due to break down of the modified constitutive equation at low temperature (1123 K) and high strain rates (≥10 s−1 ) as discussed in the previous section. In spite of this scattering, the AARE was found to be 6.74%, which reflects the good prediction capabilities of the proposed constitutive equation. 4. Conclusions Constitutive analysis of alloy D9 was carried out by performing hot compression tests in a wide range of temperatures (1123–1523 K) and strain rates (0.001–100 s−1 ). The following are the conclusions: • The influence of strain in the constitutive analysis was incorporated by considering the effect of strain on material constants (i.e. ˛, ˇ, n, Q and A). A fourth order polynomial was found suitable to represent the influence of strain on material constants with very good correlation and generalization. • The constitutive equation (considering only the compensation of strain) was found to predict flow stress precisely at strain rates of 0.1 and 1 s−1 . However, a significant deviation in the prediction is observed below 0.1 s−1 and above 1 s−1 . • The revised constitutive equation, incorporating the strain and strain rate compensation, could predict the flow stress throughout the entire temperatures and strain rates range (except at 1123 K in 10 and 100 s−1 ). The breakdown of the constitutive equation at these processing conditions is believed to be due to adiabatic temperature rise during this high strain rate deformation. Acknowledgements The authors gratefully acknowledge Dr. Baldev Raj, Director, Indira Gandhi Centre for Atomic Research, for his constant encouragement during the investigation. References [1] S.L. Mannan, P.V. Sivaprasad, in: K.H. Jurgen Buschow, R.W. Cahn, M.C. Flemings, B. Ilschner, E.J. Kramer, S. Mahajan (Eds.), Encyclopedia of Materials Science and Technology, vol. 3, Elsevier, New York, 2001, pp. 2857–2865.
S. Mandal et al. / Materials Science and Engineering A 500 (2009) 114–121 [2] P.V. Sivaprasad, Sumantra Mandal, S. Venugopal, C. Narayanan, V. Shanmugam, Baldev Raj, Transaction of the Indian Institute of Metals 59 (2006) 437–445. [3] Sumantra Mandal, P.V. Sivaprasad, S. Venugopal, K.P.N. Murthy, Baldev Raj, Materials Science and Engineering A 485 (2008) 571–580. [4] Y.-C. Lin, M.-S. Chen, J. Zhang, Materials Science and Engineering A 499 (2009) 88–92. [5] J. Jonas, C.M. Sellars, W.J.McG. Tegart, International Metallurgy Review 14 (1969) 1. [6] F.A. Slooff, J. Zhou, J. Duszczyk, L. Katgerman, Scripta Materialia 57 (2007) 759–762. [7] Z.J. Pu, K.H. Wu, J. Shai, D. Zou, Materials Science and Engineering A 192–193 (1995) 780–787.
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