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A micromechanic modeling for transformation induced plasticity in steels
Materials Science and Engineering A285 (2000) 345 – 352 www.elsevier.com/locate/msea
A micromechanic modeling for transformation induced plasticity in steels Noriyuki Tsuchida *, Y. Tomota Department of Materials Science, Faculty of Engineering, Ibaraki Uni6ersity, 4 -12 -1 Nakanarusawa, Hitachi City, Ibaraki Pref., 316 -8511, Japan
Abstract Transformation induced plasticity (TRIP) has been computed by using a micromechanic model, where the Eshelby inclusion theory, the Mori–Tanaka mean field theory and the Weng secant method are combined. This model enables us to understand experimental findings in three types of steels systematically, including (1) the influence of test temperature and strength of stress-induced martensite on elongation in metastable austenitic steels; (2) poor enhancement of elongation by TRIP of retained or precipitated austenite in Ni steels for cryogenic temperature use; and (3) the attractive combination of strength and ductility in Si bearing high strength sheet steels for automobile use that have currently been developed. A question why a small amount of austenite is effective to enhance the ductility in the last steels is clearly demonstrated by the present computations, where the effect of martensite strength for work-hardening is emphasized. © 2000 Elsevier Science S.A. All rights reserved. Keywords: Micromechanic model; Transformation induced plasticity; Steel
1. Introduction Transformation induced plasticity (TRIP) phenomenon has been discussed three times so far. The schematic illustrations of three types of TRIP steels during deformation are shown in Fig. 1. First, so called TRIP steels that show high strength and high ductility have been developed and plastic forming of metastable stainless steels have been examined [1 – 3]; type I. In type I, the microstructure is composed of austenite and martensite transformed during deformation. Such a TRIP phenomenon in austenitic steels has been studied extensively but the engineering application of the excellent ability of ductility enhancement is quite limited mainly because of high alloying, i.e. high cost. Second, the role of retained austenite for increasing toughness in 5.5– 9% Ni steels for cryogenic temperature use was discussed [4]; type II. As seen in Fig. 1 retained austenite exists in the tempered martensite matrix and transforms with stress-induced martensite, which is lath martensite. It is clear that the enhancement of elongation by TRIP is very small in type II although it is still questionable about toughness. Recently, high-Si bearing high strength sheet steels for automobile use have been developed [5]; type III. * Corresponding author.
The microstructure is that bainite exists in the ferrite matrix and retained austenite exists close to bainite as seen in Fig. 1. Retained austenite transforms with stress-induced martensite during deformation and the stress-induced martensite is lath martensite like type II steels. Because of low alloyed steels, much attention has been paid on the type III TRIP with a simple question why it shows excellent elongation in spite of containing a small amount of retained austenite like type II steels.
Fig. 1. Schematic illustration of microstructural change during deformation in three types of TRIP steels.
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2. Calculating procedures and backgrounds In this chapter, the experimental backgrounds and the procedure of the present calculations are briefly explained. Most of the experimental data used for the present calculations were taken from the previous experiments made by one of the present authors [9]. Some of them are newly added or reanalyzed.
2.1. Transformation strains in Fe–Ni–C alloys
Fig. 2. Martensitic transformation strains by cooling below Ms temperature with or without applied stress in an Fe–27Ni – 0.5C alloy. Here, ox and oy refer to transformation strain along parallel and vertical directions with respect to the applied stress.
One may consider that TRIP is ascribed to martensitic transformation strain itself. The strains appeared due to martensitic transformation were measured by using an Fe–27Ni–0.4C alloy that were austenitized at 1423 K for 3.6 ks [10]. Ms temperature of the alloy was 239 K. Tensile specimens were continuously cooled under the load related to a half of the yield strenght of the austenite at 300 K, i.e. the applied load within elastic deformation. Fig. 2 shows the transformation strains along vertical and longitudinal directions with respect to tensile direction during cooling from room temperature (RT) to 200 K with or without the applied stress. As seen in the figure, the transformation strains obtained without the applied stress show isotropic dilatation while those under the applied stress show clearly anisotropic one. The magnitude of transformation strain at 213 K along the applied stress is :1.1% where martensite volume fraction is almost 100%. This indicates that the influence of transformation strains on elongation enhanced by TRIP is small. As will be discussed later, increasing of work-hardening due to the introduction of strong martensite plays a major role for TRIP, which is different from shape memory phenomenon.
2.2. Experimental data on TRIP in Fe–Ni–C alloys
Fig. 3. Tensile properties as a function of test temperatures in an Fe– 30Ni – 0.2C alloy [9].
Understanding on TRIP phenomenon in the above three types therefore seems to be a little in confusion and then it is worth to present a systematic computed result using an appropriate model. This study presents a new model for predictions of stress – strain curve of TRIP steels — the Eshelby inclusion theory [6] and the Mori–Tanaka mean field concept [7] are employed coupled with the Weng secant method [8]. The computed results demonstrate the features of TRIP in the three types satisfactorily, particularly they reveal why enhancement of ductility is obvious in type III steels but not in type II steels.
Fig. 3 shows typical temperature dependence of elongation, yield strength and tensile strength in an Fe– 30Ni–0.2C alloy. The transformation temperatures, Ms and Md are 216 and 273 K, respectively. As is already well known, a larger elongation is obtained at a temperature between Ms and Md. Flow curves for such deformation accompanying stress-induced martensitic transformation has been studied by several workers [11–13]. Ludwigson and Berger [11] have firstly shown a kind of composite model for calculating stress–strain curves with TRIP in which plastic deformation of martensite was neglected. Such a calculation has been improved by taking the transformation strains and plastic deformation of martensite into account [12,13]. In fact, the influence of transformation strain can be found as a softening effect, obviously in the appearance of serration on a stress–strain curve and a positive temperature dependence of the yield strength in a tem-
N. Tsuchida, Y. Tomota / Materials Science and Engineering A285 (2000) 345–352
perature region near MS (see Fig. 3). The influence of transformation strains on total elongation is however not so large as is known from Fig. 2. Therefore, the transformation strain was neglected in the present calculations. For predicting the amount of uniform elongation, the most difficult point encountered in the modeling is stress partitioning between austenite and martensite. The present computation has adopted the Weng secant method [8] for stress partitioning that has successfully been used in the prediction of stress–strain curves of multi microstructure steels [14,15]. Fig. 4 shows the amount of stress-induced martensite as a function of strain at various temperatures. The local stress for inducing the martensitic transformation is increased by dislocation microstructure evolved during deformation, i.e. the applied stress is amplified. The modeling for such a stress concentration is however extremely difficult. So, as is usually done, the stress-induced transformation kinetics are described in this study as a function of overall plastic strain by the following equation proposed by Matsumura et al. [16]. Vg =
Vg 0 1+ (k/q)Vg 0(o¯ p*)q
(1)
where Vg means volume fraction of retained austenite, Vg 0 volume fraction of retained austenite before deformation, o¯ p* overall plastic strain, and k and q are constants. This equation can describe various conditions of stress-induced transformation by changing the value of k and q. The parameters in Eq. (1) were obtained in the present study by curve fitting for the data in Fig. 4. As seen, it is found that the curve at 213 K is well fitted but those of the others are not enough. As shown later, subtle difference for curve fitting for 253 K results in a big change for predicted elongation.
The stress–strain curves for austenite and martensite in a metastable austenitic steel are approximated by the experimental tensile properties obtained at 353 K (above Md) and 77 K (below Mf), respectively, where their temperature dependence was neglected. They were approximated by the Swift equation: s¯ = a(b+o¯ p*)N
(2)
where s¯ means equivalent stress and o¯ p* equivalent plastic strain. Material constants a= 883 MPa, b= 0.02, N=0.39 for austenite, and a= 2498 MPa, b= 10 − 7, N=0.29 for martensite were determined from curve fitting.
2.3. A model using the secant method Various predicting models for flow stress of multi-microstructure steels have been investigating. Uniform strain model or uniform stress model is popularly used for simple approximation. One of the present authors et al. have attempted to perform the calculation of stress– strain curves of dual microstructure steels using micromechanic modeling by employing the Eshelby inclusion theory [6] and the Mori–Tanka mean field concept [7]. But their calculated results were bigger than the measured ones [17] because the calculation model did not take the internal stress relaxation into account. The internal stresses are yielded from plastic strain difference between two constituents and are relaxed partially by dislocation motion in the vicinity of the interface. It is very important for the calculation of stress–strain curves of such steels to take the internal stress relaxation into account. Then, the Weng secant method [8] is used for the present calculations, because his method can give stress or strain partitioning after plastic relaxation mathematically. The Weng secant method is composed of the Eshelby inclusion theory, the Mori–Tanka mean field concept and the Hencky’s flow rule. In terms of internal stress relaxation, a condition for plastic deformation of the matrix is regarded as an elastic deformation where the elastic modulus changes moment by moment. Stress or strain partitioning parameter B s0 that connects stress and strain of composite with those of each component is described by the following equation [8]. B s0 =
Fig. 4. Change of the volume fraction of martensite during tensile deformation at 213, 243 and 253 K in the Fe–30Ni–0.2C alloy [9].
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b s0(m1 − m s0)+ m s0 ( f+(1− f )b s0)(m1 − m s0)+ m s0
(3)
where f refers to the volume fraction of the inclusion, b s0 is the Eshelby’s tensor for spherical inclusions; b s0 = 2(4− 5n s)/15(1 −n s) where m and n s refers to shear modulus and Poisson’ ratio. Subscripts 0, 1 and s indicate the matrix, inclusion and values for the secant moduli, respectively. The parameter B s0 changes according to deformation, so this predicting model seems to be more realistic than other models in [12,13].
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Table 1 Input data for the calculations of flow curves of three-type TRIP Type
Constituent microstructure
a (MPa)
b
N
Comment
I. Metastable austenitic steel
Austenite Martensite (stress-induced) Martensite (stress-induced)
883 2498 3750
0.02 1E-07 1E-0.7
0.39 0.29 0.29
Case 1 Case 2 (High C)
Austenitea Martensite (matrix)b Martensite (stress-induced)c
1490 1587 1587
0.02 1E-0.7 IE-0.7
0.44 0.06 0.06
Austenitea Martensite (stress-induced)b Ferriteb Bainitec
1490 6240 1003 950
0.02 1E-0.7 0.002 0.0005
0.44 0.52 0.19 0.15
II. Ni steel for cryogenic temperature use
III. Si bearing high strength sheet steel
Tempered
Examplec
a
The values are obtained by fitting for data of SUS316 steel. Concerning the details, see ref [17]. c Values for bainite change according to the volume fraction of retained austenite. b
The outline of the present calculation procedure is as follows. The calculation is divided into three stages. At first, stage I is a deformation where both the matrix and inclusion (the second phase or microstructure) are elastic. The effective bulk (k), shear (m) and Young (E) moduli of a multi-microstructure steel are given by the elastic moduli of its constituents as follows. f(k1 −k0) k = 1+ (1−f )a0(k1 −k0) +k0 k0
(4)
f(m1 −m0) m = 1+ (1− f )b0(m1 −m0) + m0 m0
(5)
E= 9km/3k + m
(6)
s¯ *B1 = a1(b1 + o¯ p*/ fB1)N1 + 3m0(1−b0)(1− f )/ f·o¯ p* (8) where o¯ p* refers to overall plastic strain which can be determined using the plastic strain of inclusion with the help of Hencky’s flow rule. Here, B0 and B1 are calculating parameters for the matrix and inclusion, respectively, and are the same with B s0, although the subscript ‘s’ is deleted. Stage III deformation is where both constituents deform plastically. The following simultaneous equations must hold. s¯ *B s1 + 3m s0(1−b s0)fB s1o p* (1) = a0(b0 + o p* (0))N0 s¯ *B − 3m (1−b )(1− f )B o * s 1
s 0
s 0
s p (1) 1
(9a)
= a1(b1 + o * )
p (1) N1
(9b)
where a0 is the Eshelby’s tensor for spherical inclusions; a0 =(1+ n)/3(1 − n). Although the elastic moduli of the matrix are assumed to be identical to those of the inclusions in this study, Eqs. (4) – (6) are needed when the secant moduli of the matrix are introduced at stage IIA and III. Stage II is an elastoplastic deformation stage. If the matrix yield first, stage IIA would be calculated and the equivalent stress would be provided by using the constitutive equation of the matrix at a given plastic strain, as follows:
Here o p*(1) refers to plastic strain in the inclusion and b is the Eshelby tensor under the Weng secant method. The input data for a, b and N in Eq. (2) used for the present calculations are tabulated in Table 1. The other input data required are the elastic moduli and the volume fraction of the constituents. The following plastic instability condition is employed for predicting the uniform elongation and tensile strength.
s¯ = a0(b0 +o p* (0))N0/B s0
ds¯ * =s¯ * do¯*
(7)
where a calculating parameter, B s0 , is associated with the stress or strain partition under the Weng secant method. To determine B s0, we can use Eqs. (4)–(6) by inserting the secant moduli (m s0, k s0) instead of the elastic moduli of the matrix (m0, k0). If the inclusion yields first, stage IIB can take place so that we can use the original Eshelby’s inclusion theory coupled with the Mori – Tanaka’s mean field concept [17]. In this case, the flow stress is given by
s 0
(10)
In the case of Ni steels for cryogenic temperature use and Si-bearing high strength sheet steels, numbers of constituent microstructures are more than three. Here, we can adopt the iterative calculating procedure of the secant method proposed by Rudiono and Tomota [15] for such cases. Thus, we can calculate overall stress– strain curves of three types of TRIP steels using the Weng secant method, the database for single microstructure steels [14] and the stress-induced transformation equation proposed by Matsumura et al. [16].
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Fig. 5. Comparisons between the calculated true stresses vs. true strain curves and the measured ones [9] at (a) 213; (b) 243; and (c) 253 K in the Fe– 30Ni – 0.2C alloy, where work-hardening rates are also shown.
3. Computed results and discussions
optimum transformation rate with deformation for obtaining the largest elongation.
3.1. Type I TRIP in metastable austenitic steels The stress–strain curves of the Fe – 30Ni – 0.2C alloy at 213, 243 and 253 K are presented in Fig. 5 (a)–(c), respectively. First of all, the enhancement of elongation at a temperature between Ms and Md and the overall flow curves are found to be excellently simulated when the predictions are qualitatively compared with measured flow curves. When the kinetics of stress-induced transformation can precisely be fitted by Eq. (1) in Fig. 5, say, at 213 K, the calculated tensile strength and uniform elongation (square symbols in Fig. 5) agree well with measured one (solid square ones). Such a quantitative agreement is however not so good in Fig. 5(b) and (c), i.e. for example, the maximum volume fraction of martensite by the transformation seems to be saturated at about 30% at 253 K, but the volume fraction continues to increase in the fitted curve by Eq. (1). Therefore, if the transformation kinetics is more reasonably described, agreements between calculated uniform elongations and measured ones would become better. It should be noted here that the transformation must be progressed little by little during deformation in order to obtain a larger elongation; there is an optimum rate for the transformation that leads to increase of work-hardening at a later stage of deformation. Fig. 6 shows a calculated example where the strength of stress-induced martensite is assumed to become one and half times larger than that for Fig. 5 in tensile strength. As seen, a larger elongation is found to appear; this is confirmed by the experiments on the influence of carbon content on elongation, where the elongation of more than 200% has been obtained by adding 0.4–0.6 mass percent carbon, i.e. by increasing the strength of stress-induced martensite [18]. It was also confirmed from calculations for various conditions that higher the strength of martensite, slower is the
3.2. Type II TRIP in retained or precipitated austenite The microstructure of Ni bearing steels for cryogenic temperature use is composed of tempered martensite and austenite. The influence of austenite stability on ductility, toughness and fatigue properties were extensively discussed [4]. As long as discussion is limited to elongation obtained by tensile testing, the influence of stress-induced martensitic transformation is almost negligible.
Fig. 6. Effect of the strength of stress-induced martensite on calculated true stress vs. true strain curves and work-hardening rate vs. true strain curves; high carbon martensite, i.e. high strength, is assumed in case 2.
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the imaginary inclusion (austenite and stress-induced martensite mixture) is calculated by repeating the Weng secant method. Fig. 8 shows the calculated stress–strain curves with or without TRIP in the cases that volume fraction of retained austenite is assumed to be 5 or 10%. As seen, TRIP effect on elongation is very small, for instance only : 0.5% in the case of 10% retained austenite. Thus, the calculated results agree well with experimental data in Fig. 8. Therefore, in the case of type II where the strength of stress-induced martensite is comparable with that of the matrix, a larger volume fraction of the retained austenite is required in order to obtain a larger elongation by TRIP. Fig. 7. Effect of the volume fraction of pre-existing martensite on the elongation with or without TRIP in a Fe–25Ni–0.4C alloy [19].
Fig. 8. Calculated true stress vs. true strain curves to examine TRIP effect for type II.
Fig. 7 shows the experimental data on the influence of the volume fraction of pre-existing martensite on elongation in an Fe – 25Ni – 0.4C alloy [19]. With decreasing volume fraction of retained austenite, the enhancement of elongation by TRIP becomes smaller. The enhancement is of negligible order in the case of several volume percent of retained austenite that is almost the maximum volume fraction for the above Ni steels. Such a case can be simulated by the above method, where the starting microstructure is already composed of martensite and austenite. The calculation is performed by using the iterative calculating procedure as mentioned above. The stress – strain curve for an austenite and stress-induced martensite mixture is calculated first by the Weng secant method as an imaginary inclusion. And then a stress – strain curve for an alloy, which is composed of tempered martensite and
3.3. Type III TRIP in high strength Si bearing sheet steels The steels are artificially designed for utilizing TRIP effect without adding a large amount of alloying elements. Approximately 2Si, 1.5Mn and 0.14C (in mass percent) are main chemical compositions. The steels are subjected to controlled cooling to result in showing a ferrite–bainite–austenite structure. Since carbide formation is strongly suppressed by Si adding during bainitic transformation, carbon-enriched austenite can be retained and it transforms to martensite when deformed plastically. The carbon concentration in austenite depends on process conditions and is considered to be near 1.0 mass percent. Therefore, the strength of martensite formed from such high carbon austenite must be much stronger than that of the matrix (ferrite and bainite mixture), that is different from the case of type II. For these steels, a computing modeling has been given by Geol et al. [20] with the assumption that strain or stress partitioning between the constituents was given by a fixed ratio (a fixed slope in a stress– strain diagram). They did not give reasonable interpretation for adopting the slope except the best fitting between calculations and experiments. The present model may be able to provide such a partitioning, a priori. The calculations in type III have been performed by the following order; (1) calculation of a stress–strain curve of ferrite and bainite mixture as an imaginary matrix; (2) calculation of a stress–strain curve of a retained austenite and stress-induced martensite mixture as an imaginary inclusion; and (3) calculation of a stress–strain curve whose constituents are composed of the imaginary matrix and inclusion. The calculated stress–strain curves of type III are shown in Fig. 9. We changed the value of q in the equation of stress-induced transformation (Eq. (1)) to examine the influence of the transformation rate in type III. As seen, the bigger the value of q becomes, the larger enhancement of elongation is obtained by TRIP. So there is an optimum rate of the stress-induced martensitic transformation in the
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Si bearing high strength steels similarly to metastable austenitic steels. On the other hand, the enhancement of elongation by TRIP in Type III is about 4% as shown in Fig. 9, which is about eight times larger than that in type II (note the scale of abscissa in the figure). The reason for such influence is that the strength of the stress-induced martensite is much higher than that of ferrite–bainite matrix. As listed in Table 1, the tensile strength of martensite of type III is about two and half times as strong as that of type II. It is therefore possible to obtain a larger elongation by increasing the amount
351
of carbon in austenite from the practical point of view. It is also expected to obtain a larger elongation by increasing the amount of retained austenite. The difference between type II and type III in terms of the enhancement of elongation by TRIP can therefore be successfully demonstrated by the present calculations.
4. Summary A new modeling of TRIP phenomenon is made, where the Eshelby inclusion theory, the Mori–Tanaka mean field concept and the Weng secant method are combined. The model is concluded to show reasonable predictions of TRIP behavior in three-type steels. The comparisons among these types are summarized in Fig. 10, in which the difference between type II and type III is ascribed to the strength of stress-induced martensite. In order to obtain a larger elongation, the stress-induced transformation should occur to suppress the start of necking effectively. Such an optimum transformation rate can be obtained at a temperature between Ms and Md, depending on the strength of martensite. Based on the excellent agreements between predictions and experimental data, the present model must be powerful for alloy design.
Acknowledgements Fig. 9. Calculated true stress vs. true stress curves to examine TRIP effect for type III where the influence of martensitic transformation rate as a function of strain, i.e. q in Eq. (1) is shown.
We are grateful to Drs K Nagai and K. Tsuzaki for helpful discussions and advices. This work is financially supported by the special coordination fund of the Science and Technology Agency, Japan.
References
Fig. 10. Summary of calculated results for three types of TRIP steel in terms of the enhancement of elongation by TRIP.
[1] V.F. Zackay, E.R. Parker, D. Fahr, R. Bush, ASM Trans. Q. 60 (1967) 252. [2] I. Tamura, Met. Sci. 16 (1982) 245. [3] T. Maki, Ph.D. Thesis, Kyoto University, 1973. [4] K. Nagai, K. Shibata, T. Fujita, Y. Ujiie, Trans. ISIJ 22 (1982) 696. [5] O. Matsumura, Y. Sakuma, H. Takechi, Trans. ISIJ 27 (1987) 570. [6] J.D. Eshelby, Proc. R. Soc. Lond. A241 (1957) 371. [7] T. Mori, K. Tanaka, Acta Met. 21 (1973) 571. [8] G.J. Weng, J. Mech. Phys. Solids 38 (1990) 419. [9] Y. Tomota, Thesis for Masters Degree, Kyoto University, 1972. [10] K. Tanabe, Y. Tomota, K. Saga, K. Kuroki, Bull Faculty of Engineering, Ibaraki University, 27 (1979) 15. [11] D.C. Ludwigson, J.A. Berger, J. Iron Steel Inst. 207 (1969) 63. [12] R.G. Stringfello, D.M. Parks, G.B. Olson, Acta Mater. 40 (1992) 1703. [13] N. Tsutta, T. Iwamoto, J. Jpn. STP 37 (1996) 492. [14] Y. Tomota, T. Ohnuki, M. Umemoto, Proc. Risø Int. Symp. Mater. Sci., Roskilde, 1994, pp. 59.
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[15] Rudiono, Y. Tomota, Acta Mater. 45 (5) (1997) 1923. [16] O. Matsumura, Y. Sakuma, H. Takechi, Scr. Met. 21 (1987) 301. [17] Y. Tomota, K. Kuroki, T. Mori, I. Tamura, Mater. Sci. Eng. 24 (1973) 85. [18] I. Tamura, T. Maki, S. Shimooka, M. Okada, Y. Tomota, Proc.
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Conf. High Strength Martensitic Steels, Havirov, Czechoslovakia, 1972, pp. 118. [19] I. Tamura, Y. Tomota, M. Ozawa, Proc. 3rd Int. Conf. Strength of Metals and Alloys, Institute of Metals, Cambridge, 1973, p. 611. [20] N.C. Geol, S. Sangal, K. Tangri, Met. Trans. 16A (1985) 2013.
A micromechanic modeling for transformation induced plasticity in steels Noriyuki Tsuchida *, Y. Tomota Department of Materials Science, Faculty of Engineering, Ibaraki Uni6ersity, 4 -12 -1 Nakanarusawa, Hitachi City, Ibaraki Pref., 316 -8511, Japan
Abstract Transformation induced plasticity (TRIP) has been computed by using a micromechanic model, where the Eshelby inclusion theory, the Mori–Tanaka mean field theory and the Weng secant method are combined. This model enables us to understand experimental findings in three types of steels systematically, including (1) the influence of test temperature and strength of stress-induced martensite on elongation in metastable austenitic steels; (2) poor enhancement of elongation by TRIP of retained or precipitated austenite in Ni steels for cryogenic temperature use; and (3) the attractive combination of strength and ductility in Si bearing high strength sheet steels for automobile use that have currently been developed. A question why a small amount of austenite is effective to enhance the ductility in the last steels is clearly demonstrated by the present computations, where the effect of martensite strength for work-hardening is emphasized. © 2000 Elsevier Science S.A. All rights reserved. Keywords: Micromechanic model; Transformation induced plasticity; Steel
1. Introduction Transformation induced plasticity (TRIP) phenomenon has been discussed three times so far. The schematic illustrations of three types of TRIP steels during deformation are shown in Fig. 1. First, so called TRIP steels that show high strength and high ductility have been developed and plastic forming of metastable stainless steels have been examined [1 – 3]; type I. In type I, the microstructure is composed of austenite and martensite transformed during deformation. Such a TRIP phenomenon in austenitic steels has been studied extensively but the engineering application of the excellent ability of ductility enhancement is quite limited mainly because of high alloying, i.e. high cost. Second, the role of retained austenite for increasing toughness in 5.5– 9% Ni steels for cryogenic temperature use was discussed [4]; type II. As seen in Fig. 1 retained austenite exists in the tempered martensite matrix and transforms with stress-induced martensite, which is lath martensite. It is clear that the enhancement of elongation by TRIP is very small in type II although it is still questionable about toughness. Recently, high-Si bearing high strength sheet steels for automobile use have been developed [5]; type III. * Corresponding author.
The microstructure is that bainite exists in the ferrite matrix and retained austenite exists close to bainite as seen in Fig. 1. Retained austenite transforms with stress-induced martensite during deformation and the stress-induced martensite is lath martensite like type II steels. Because of low alloyed steels, much attention has been paid on the type III TRIP with a simple question why it shows excellent elongation in spite of containing a small amount of retained austenite like type II steels.
Fig. 1. Schematic illustration of microstructural change during deformation in three types of TRIP steels.
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2. Calculating procedures and backgrounds In this chapter, the experimental backgrounds and the procedure of the present calculations are briefly explained. Most of the experimental data used for the present calculations were taken from the previous experiments made by one of the present authors [9]. Some of them are newly added or reanalyzed.
2.1. Transformation strains in Fe–Ni–C alloys
Fig. 2. Martensitic transformation strains by cooling below Ms temperature with or without applied stress in an Fe–27Ni – 0.5C alloy. Here, ox and oy refer to transformation strain along parallel and vertical directions with respect to the applied stress.
One may consider that TRIP is ascribed to martensitic transformation strain itself. The strains appeared due to martensitic transformation were measured by using an Fe–27Ni–0.4C alloy that were austenitized at 1423 K for 3.6 ks [10]. Ms temperature of the alloy was 239 K. Tensile specimens were continuously cooled under the load related to a half of the yield strenght of the austenite at 300 K, i.e. the applied load within elastic deformation. Fig. 2 shows the transformation strains along vertical and longitudinal directions with respect to tensile direction during cooling from room temperature (RT) to 200 K with or without the applied stress. As seen in the figure, the transformation strains obtained without the applied stress show isotropic dilatation while those under the applied stress show clearly anisotropic one. The magnitude of transformation strain at 213 K along the applied stress is :1.1% where martensite volume fraction is almost 100%. This indicates that the influence of transformation strains on elongation enhanced by TRIP is small. As will be discussed later, increasing of work-hardening due to the introduction of strong martensite plays a major role for TRIP, which is different from shape memory phenomenon.
2.2. Experimental data on TRIP in Fe–Ni–C alloys
Fig. 3. Tensile properties as a function of test temperatures in an Fe– 30Ni – 0.2C alloy [9].
Understanding on TRIP phenomenon in the above three types therefore seems to be a little in confusion and then it is worth to present a systematic computed result using an appropriate model. This study presents a new model for predictions of stress – strain curve of TRIP steels — the Eshelby inclusion theory [6] and the Mori–Tanaka mean field concept [7] are employed coupled with the Weng secant method [8]. The computed results demonstrate the features of TRIP in the three types satisfactorily, particularly they reveal why enhancement of ductility is obvious in type III steels but not in type II steels.
Fig. 3 shows typical temperature dependence of elongation, yield strength and tensile strength in an Fe– 30Ni–0.2C alloy. The transformation temperatures, Ms and Md are 216 and 273 K, respectively. As is already well known, a larger elongation is obtained at a temperature between Ms and Md. Flow curves for such deformation accompanying stress-induced martensitic transformation has been studied by several workers [11–13]. Ludwigson and Berger [11] have firstly shown a kind of composite model for calculating stress–strain curves with TRIP in which plastic deformation of martensite was neglected. Such a calculation has been improved by taking the transformation strains and plastic deformation of martensite into account [12,13]. In fact, the influence of transformation strain can be found as a softening effect, obviously in the appearance of serration on a stress–strain curve and a positive temperature dependence of the yield strength in a tem-
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perature region near MS (see Fig. 3). The influence of transformation strains on total elongation is however not so large as is known from Fig. 2. Therefore, the transformation strain was neglected in the present calculations. For predicting the amount of uniform elongation, the most difficult point encountered in the modeling is stress partitioning between austenite and martensite. The present computation has adopted the Weng secant method [8] for stress partitioning that has successfully been used in the prediction of stress–strain curves of multi microstructure steels [14,15]. Fig. 4 shows the amount of stress-induced martensite as a function of strain at various temperatures. The local stress for inducing the martensitic transformation is increased by dislocation microstructure evolved during deformation, i.e. the applied stress is amplified. The modeling for such a stress concentration is however extremely difficult. So, as is usually done, the stress-induced transformation kinetics are described in this study as a function of overall plastic strain by the following equation proposed by Matsumura et al. [16]. Vg =
Vg 0 1+ (k/q)Vg 0(o¯ p*)q
(1)
where Vg means volume fraction of retained austenite, Vg 0 volume fraction of retained austenite before deformation, o¯ p* overall plastic strain, and k and q are constants. This equation can describe various conditions of stress-induced transformation by changing the value of k and q. The parameters in Eq. (1) were obtained in the present study by curve fitting for the data in Fig. 4. As seen, it is found that the curve at 213 K is well fitted but those of the others are not enough. As shown later, subtle difference for curve fitting for 253 K results in a big change for predicted elongation.
The stress–strain curves for austenite and martensite in a metastable austenitic steel are approximated by the experimental tensile properties obtained at 353 K (above Md) and 77 K (below Mf), respectively, where their temperature dependence was neglected. They were approximated by the Swift equation: s¯ = a(b+o¯ p*)N
(2)
where s¯ means equivalent stress and o¯ p* equivalent plastic strain. Material constants a= 883 MPa, b= 0.02, N=0.39 for austenite, and a= 2498 MPa, b= 10 − 7, N=0.29 for martensite were determined from curve fitting.
2.3. A model using the secant method Various predicting models for flow stress of multi-microstructure steels have been investigating. Uniform strain model or uniform stress model is popularly used for simple approximation. One of the present authors et al. have attempted to perform the calculation of stress– strain curves of dual microstructure steels using micromechanic modeling by employing the Eshelby inclusion theory [6] and the Mori–Tanka mean field concept [7]. But their calculated results were bigger than the measured ones [17] because the calculation model did not take the internal stress relaxation into account. The internal stresses are yielded from plastic strain difference between two constituents and are relaxed partially by dislocation motion in the vicinity of the interface. It is very important for the calculation of stress–strain curves of such steels to take the internal stress relaxation into account. Then, the Weng secant method [8] is used for the present calculations, because his method can give stress or strain partitioning after plastic relaxation mathematically. The Weng secant method is composed of the Eshelby inclusion theory, the Mori–Tanka mean field concept and the Hencky’s flow rule. In terms of internal stress relaxation, a condition for plastic deformation of the matrix is regarded as an elastic deformation where the elastic modulus changes moment by moment. Stress or strain partitioning parameter B s0 that connects stress and strain of composite with those of each component is described by the following equation [8]. B s0 =
Fig. 4. Change of the volume fraction of martensite during tensile deformation at 213, 243 and 253 K in the Fe–30Ni–0.2C alloy [9].
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b s0(m1 − m s0)+ m s0 ( f+(1− f )b s0)(m1 − m s0)+ m s0
(3)
where f refers to the volume fraction of the inclusion, b s0 is the Eshelby’s tensor for spherical inclusions; b s0 = 2(4− 5n s)/15(1 −n s) where m and n s refers to shear modulus and Poisson’ ratio. Subscripts 0, 1 and s indicate the matrix, inclusion and values for the secant moduli, respectively. The parameter B s0 changes according to deformation, so this predicting model seems to be more realistic than other models in [12,13].
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Table 1 Input data for the calculations of flow curves of three-type TRIP Type
Constituent microstructure
a (MPa)
b
N
Comment
I. Metastable austenitic steel
Austenite Martensite (stress-induced) Martensite (stress-induced)
883 2498 3750
0.02 1E-07 1E-0.7
0.39 0.29 0.29
Case 1 Case 2 (High C)
Austenitea Martensite (matrix)b Martensite (stress-induced)c
1490 1587 1587
0.02 1E-0.7 IE-0.7
0.44 0.06 0.06
Austenitea Martensite (stress-induced)b Ferriteb Bainitec
1490 6240 1003 950
0.02 1E-0.7 0.002 0.0005
0.44 0.52 0.19 0.15
II. Ni steel for cryogenic temperature use
III. Si bearing high strength sheet steel
Tempered
Examplec
a
The values are obtained by fitting for data of SUS316 steel. Concerning the details, see ref [17]. c Values for bainite change according to the volume fraction of retained austenite. b
The outline of the present calculation procedure is as follows. The calculation is divided into three stages. At first, stage I is a deformation where both the matrix and inclusion (the second phase or microstructure) are elastic. The effective bulk (k), shear (m) and Young (E) moduli of a multi-microstructure steel are given by the elastic moduli of its constituents as follows. f(k1 −k0) k = 1+ (1−f )a0(k1 −k0) +k0 k0
(4)
f(m1 −m0) m = 1+ (1− f )b0(m1 −m0) + m0 m0
(5)
E= 9km/3k + m
(6)
s¯ *B1 = a1(b1 + o¯ p*/ fB1)N1 + 3m0(1−b0)(1− f )/ f·o¯ p* (8) where o¯ p* refers to overall plastic strain which can be determined using the plastic strain of inclusion with the help of Hencky’s flow rule. Here, B0 and B1 are calculating parameters for the matrix and inclusion, respectively, and are the same with B s0, although the subscript ‘s’ is deleted. Stage III deformation is where both constituents deform plastically. The following simultaneous equations must hold. s¯ *B s1 + 3m s0(1−b s0)fB s1o p* (1) = a0(b0 + o p* (0))N0 s¯ *B − 3m (1−b )(1− f )B o * s 1
s 0
s 0
s p (1) 1
(9a)
= a1(b1 + o * )
p (1) N1
(9b)
where a0 is the Eshelby’s tensor for spherical inclusions; a0 =(1+ n)/3(1 − n). Although the elastic moduli of the matrix are assumed to be identical to those of the inclusions in this study, Eqs. (4) – (6) are needed when the secant moduli of the matrix are introduced at stage IIA and III. Stage II is an elastoplastic deformation stage. If the matrix yield first, stage IIA would be calculated and the equivalent stress would be provided by using the constitutive equation of the matrix at a given plastic strain, as follows:
Here o p*(1) refers to plastic strain in the inclusion and b is the Eshelby tensor under the Weng secant method. The input data for a, b and N in Eq. (2) used for the present calculations are tabulated in Table 1. The other input data required are the elastic moduli and the volume fraction of the constituents. The following plastic instability condition is employed for predicting the uniform elongation and tensile strength.
s¯ = a0(b0 +o p* (0))N0/B s0
ds¯ * =s¯ * do¯*
(7)
where a calculating parameter, B s0 , is associated with the stress or strain partition under the Weng secant method. To determine B s0, we can use Eqs. (4)–(6) by inserting the secant moduli (m s0, k s0) instead of the elastic moduli of the matrix (m0, k0). If the inclusion yields first, stage IIB can take place so that we can use the original Eshelby’s inclusion theory coupled with the Mori – Tanaka’s mean field concept [17]. In this case, the flow stress is given by
s 0
(10)
In the case of Ni steels for cryogenic temperature use and Si-bearing high strength sheet steels, numbers of constituent microstructures are more than three. Here, we can adopt the iterative calculating procedure of the secant method proposed by Rudiono and Tomota [15] for such cases. Thus, we can calculate overall stress– strain curves of three types of TRIP steels using the Weng secant method, the database for single microstructure steels [14] and the stress-induced transformation equation proposed by Matsumura et al. [16].
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Fig. 5. Comparisons between the calculated true stresses vs. true strain curves and the measured ones [9] at (a) 213; (b) 243; and (c) 253 K in the Fe– 30Ni – 0.2C alloy, where work-hardening rates are also shown.
3. Computed results and discussions
optimum transformation rate with deformation for obtaining the largest elongation.
3.1. Type I TRIP in metastable austenitic steels The stress–strain curves of the Fe – 30Ni – 0.2C alloy at 213, 243 and 253 K are presented in Fig. 5 (a)–(c), respectively. First of all, the enhancement of elongation at a temperature between Ms and Md and the overall flow curves are found to be excellently simulated when the predictions are qualitatively compared with measured flow curves. When the kinetics of stress-induced transformation can precisely be fitted by Eq. (1) in Fig. 5, say, at 213 K, the calculated tensile strength and uniform elongation (square symbols in Fig. 5) agree well with measured one (solid square ones). Such a quantitative agreement is however not so good in Fig. 5(b) and (c), i.e. for example, the maximum volume fraction of martensite by the transformation seems to be saturated at about 30% at 253 K, but the volume fraction continues to increase in the fitted curve by Eq. (1). Therefore, if the transformation kinetics is more reasonably described, agreements between calculated uniform elongations and measured ones would become better. It should be noted here that the transformation must be progressed little by little during deformation in order to obtain a larger elongation; there is an optimum rate for the transformation that leads to increase of work-hardening at a later stage of deformation. Fig. 6 shows a calculated example where the strength of stress-induced martensite is assumed to become one and half times larger than that for Fig. 5 in tensile strength. As seen, a larger elongation is found to appear; this is confirmed by the experiments on the influence of carbon content on elongation, where the elongation of more than 200% has been obtained by adding 0.4–0.6 mass percent carbon, i.e. by increasing the strength of stress-induced martensite [18]. It was also confirmed from calculations for various conditions that higher the strength of martensite, slower is the
3.2. Type II TRIP in retained or precipitated austenite The microstructure of Ni bearing steels for cryogenic temperature use is composed of tempered martensite and austenite. The influence of austenite stability on ductility, toughness and fatigue properties were extensively discussed [4]. As long as discussion is limited to elongation obtained by tensile testing, the influence of stress-induced martensitic transformation is almost negligible.
Fig. 6. Effect of the strength of stress-induced martensite on calculated true stress vs. true strain curves and work-hardening rate vs. true strain curves; high carbon martensite, i.e. high strength, is assumed in case 2.
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the imaginary inclusion (austenite and stress-induced martensite mixture) is calculated by repeating the Weng secant method. Fig. 8 shows the calculated stress–strain curves with or without TRIP in the cases that volume fraction of retained austenite is assumed to be 5 or 10%. As seen, TRIP effect on elongation is very small, for instance only : 0.5% in the case of 10% retained austenite. Thus, the calculated results agree well with experimental data in Fig. 8. Therefore, in the case of type II where the strength of stress-induced martensite is comparable with that of the matrix, a larger volume fraction of the retained austenite is required in order to obtain a larger elongation by TRIP. Fig. 7. Effect of the volume fraction of pre-existing martensite on the elongation with or without TRIP in a Fe–25Ni–0.4C alloy [19].
Fig. 8. Calculated true stress vs. true strain curves to examine TRIP effect for type II.
Fig. 7 shows the experimental data on the influence of the volume fraction of pre-existing martensite on elongation in an Fe – 25Ni – 0.4C alloy [19]. With decreasing volume fraction of retained austenite, the enhancement of elongation by TRIP becomes smaller. The enhancement is of negligible order in the case of several volume percent of retained austenite that is almost the maximum volume fraction for the above Ni steels. Such a case can be simulated by the above method, where the starting microstructure is already composed of martensite and austenite. The calculation is performed by using the iterative calculating procedure as mentioned above. The stress – strain curve for an austenite and stress-induced martensite mixture is calculated first by the Weng secant method as an imaginary inclusion. And then a stress – strain curve for an alloy, which is composed of tempered martensite and
3.3. Type III TRIP in high strength Si bearing sheet steels The steels are artificially designed for utilizing TRIP effect without adding a large amount of alloying elements. Approximately 2Si, 1.5Mn and 0.14C (in mass percent) are main chemical compositions. The steels are subjected to controlled cooling to result in showing a ferrite–bainite–austenite structure. Since carbide formation is strongly suppressed by Si adding during bainitic transformation, carbon-enriched austenite can be retained and it transforms to martensite when deformed plastically. The carbon concentration in austenite depends on process conditions and is considered to be near 1.0 mass percent. Therefore, the strength of martensite formed from such high carbon austenite must be much stronger than that of the matrix (ferrite and bainite mixture), that is different from the case of type II. For these steels, a computing modeling has been given by Geol et al. [20] with the assumption that strain or stress partitioning between the constituents was given by a fixed ratio (a fixed slope in a stress– strain diagram). They did not give reasonable interpretation for adopting the slope except the best fitting between calculations and experiments. The present model may be able to provide such a partitioning, a priori. The calculations in type III have been performed by the following order; (1) calculation of a stress–strain curve of ferrite and bainite mixture as an imaginary matrix; (2) calculation of a stress–strain curve of a retained austenite and stress-induced martensite mixture as an imaginary inclusion; and (3) calculation of a stress–strain curve whose constituents are composed of the imaginary matrix and inclusion. The calculated stress–strain curves of type III are shown in Fig. 9. We changed the value of q in the equation of stress-induced transformation (Eq. (1)) to examine the influence of the transformation rate in type III. As seen, the bigger the value of q becomes, the larger enhancement of elongation is obtained by TRIP. So there is an optimum rate of the stress-induced martensitic transformation in the
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Si bearing high strength steels similarly to metastable austenitic steels. On the other hand, the enhancement of elongation by TRIP in Type III is about 4% as shown in Fig. 9, which is about eight times larger than that in type II (note the scale of abscissa in the figure). The reason for such influence is that the strength of the stress-induced martensite is much higher than that of ferrite–bainite matrix. As listed in Table 1, the tensile strength of martensite of type III is about two and half times as strong as that of type II. It is therefore possible to obtain a larger elongation by increasing the amount
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of carbon in austenite from the practical point of view. It is also expected to obtain a larger elongation by increasing the amount of retained austenite. The difference between type II and type III in terms of the enhancement of elongation by TRIP can therefore be successfully demonstrated by the present calculations.
4. Summary A new modeling of TRIP phenomenon is made, where the Eshelby inclusion theory, the Mori–Tanaka mean field concept and the Weng secant method are combined. The model is concluded to show reasonable predictions of TRIP behavior in three-type steels. The comparisons among these types are summarized in Fig. 10, in which the difference between type II and type III is ascribed to the strength of stress-induced martensite. In order to obtain a larger elongation, the stress-induced transformation should occur to suppress the start of necking effectively. Such an optimum transformation rate can be obtained at a temperature between Ms and Md, depending on the strength of martensite. Based on the excellent agreements between predictions and experimental data, the present model must be powerful for alloy design.
Acknowledgements Fig. 9. Calculated true stress vs. true stress curves to examine TRIP effect for type III where the influence of martensitic transformation rate as a function of strain, i.e. q in Eq. (1) is shown.
We are grateful to Drs K Nagai and K. Tsuzaki for helpful discussions and advices. This work is financially supported by the special coordination fund of the Science and Technology Agency, Japan.
References
Fig. 10. Summary of calculated results for three types of TRIP steel in terms of the enhancement of elongation by TRIP.
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