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Mechanical behavior of tempered martensite: Characterization and modeling
Materials Science & Engineering A 706 (2017) 38–47
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Mechanical behavior of tempered martensite: Characterization and modeling L.R. Cupertino Malheirosa,1, E.A. Pachon Rodrigueza, A. Arlazarova, a
MARK
⁎
ArcelorMittal Global Research and Development, Voie Romaine BP30320, 57283 Maizières-lès-Metz Cedex, France
A R T I C L E I N F O
A B S T R A C T
Keywords: Martensite Tempering Strain hardening Modeling Mechanical behavior
Mechanical characterization of five low-carbon martensitic steels tempered at a wide range of temperatures and times was performed. Analysis of the relationships between mechanical properties, hardness and tempering conditions was completed. Microstructural observations revealed the precipitation of carbides and consequent decrease of carbon in solid solution with tempering. To describe the mechanical behavior of tempered martensite, a simplified Continuous Composite model, reported previously, was adapted. A good agreement between the model and the experimental stress-strain curves was achieved with only one fitting parameter. Further directions for improvement of the model were also proposed.
1. Introduction
The precipitation of rod-shaped cementite starts at 250 °C and it tends to be replaced by spheroidal particles at 400 °C. The recovery of the very deformed martensitic structure occurs more significantly at temperatures above 400 °C. Finally, recrystallization and grain growth are the last stages of tempering. Caron and Krauss [11] also studied the changes of the lath martensite when tempering at high temperatures. They noticed that the elongated packet-lath morphology of martensite is maintained up to long times at high tempering temperatures. In addition, they observed the laths coarsening using electron microscopy. In another work, Swarr and Krauss [12] observed the presence of discshaped and spherical carbides with 10 nm in diameter within the lattices, and larger carbides – 20 nm in thickness and 100 nm in length - in the lath boundaries of the tempered steel. They also observed the transformations in the substructure of the as-quenched and tempered martensitic steels before and after being deformed by tensile testing. While the as-quenched sample has a cellular structure that develops during deformation, the tempered sample has a uniform distribution of dislocations that practically does not change during deformation. Tempering also impacts hardness. Grange et al. [13] observed that the hardness of as-quenched martensite depends only on the carbon content. However, the hardness of tempered martensite substantially increases with alloying elements additions. Hollomon and Jaffe [14] modeled the hardness evolution with tempering temperatures and times. They found the following relationship between the tempering conditions and the hardness:
The use of hardened steels date from thousands of years ago [1]. However, characterization and understanding of their microstructures started only after the availability of analytical and testing techniques in the end of the nineteenth century. The name martensite was given to honor the important contribution of Adolf Martens [2] in the metallographic observations of the hard phase found in the quenched steels. Now, this name is used to designate any phase in ferrous and nonferrous systems formed by diffusionless and solid-state shear transformation. In steels, martensite is a very complex phase that was studied in the twentieth century by many researchers [3–8]. Its structure varies from lath-like to plate-like, its substructure from dislocations to twins and its crystallography from BCC to BCT when the carbon content increases. Tempering process is the reheating of a martensitic steel to a certain temperature, accompanied with the holding for a certain time, in order to improve the toughness and strength-ductility balance of the martensitic product. Tempered martensite has received increasing attention since it is one of the phases present in most of advanced high strength steels (AHSS). Speich and Leslie [9,10] gave an account of the different microstructural changes that occur during tempering. Firstly, at tempering temperatures between 100 and 200 °C, the carbon segregation to the defects, started during quenching, continues to occur. Therefore, there is a depletion of carbon in the solid solution inhibiting the ε-carbide precipitation in low carbon steels. Secondly, at temperatures between 200 and 300 °C, the decomposition of retained austenite usually begins.
⁎
1
R c = Hc −0.00457T (13+logt )
Corresponding author. E-mail address: [email protected] (A. Arlazarov). Present address: Department of Metallurgical and Materials Engineering, Universidade Federal de Minas Gerais, Belo Horizonte, MG 31270-901, Brazil.
http://dx.doi.org/10.1016/j.msea.2017.08.089 Received 15 May 2017; Received in revised form 10 August 2017; Accepted 25 August 2017 Available online 26 August 2017 0921-5093/ © 2017 Elsevier B.V. All rights reserved.
(1)
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where Rc is the Rockwell C hardness, Hc is a characteristic hardness, T is the tempering temperature in Kelvin and t is the tempering time in seconds. Despite all the work performed on tempered martensite [15–21], the microstructural evolution during tempering at a fine scale and the effects of different microstructural changes on the strain-hardening of tempered martensite are still unclear. In this context, the purpose of this work is to propose a composite model for the prediction of the stress-strain curves of tempered martensitic steels. Composite models started to be applied in the 20th century. They are based on the Masing's model of polycrystal plasticity [22], in which the individual grains yield at different external stress values. Pedersen et al. [23] proposed a composite model to explain the Bauschinger effect of workhardened copper. In this model the dislocations tangles act as hard but deformable particles in a soft phase, practically free of dislocations. Mughrabi [24] proposed a composite model for the plasticity of crystals with cell structures. In his model, the cells walls are considered as hard phases separated by soft regions with low dislocation density. Polák et al. [25,26] built a more sophisticated composite model. Instead of considering just two phases as in the models of Pederson and Mughrabi, i.e., one phase with high and another with low dislocation density, this model deals with a continuous distribution of volumes with different internal critical flow stresses (σic). This distribution was characterized by a probability density function that can be written as:
∫0
∞
Fig. 1. Experimental true stress-true strain curves of some tempering conditions performed with the HS steel.
ISO procedure. The presence, size and morphology of cementite was observed with SEM after etching with Picral reagent (4 g of picric acid in 100 ml of ethanol). Finally, conventional Vickers micro-hardness measurements were performed at 13 locations of the polished surfaces using the load of 1 kg and holding time of 10 s.
3. Results
f (σic ) dσic = 1
(2) Fig. 1 shows some of the experimental tensile curves obtained in the present study for the HS steel. These curves illustrate the mechanical behavior evolution with tempering and the expected softening of the steel. Similarly, Fig. 2 presents some of the experimental tensile curves for the other steels (LC, HCHS, HA and LM). The softening behavior observed in Fig. 1 also occurs for these other steels. However, the LC steel presented some sort of temper embrittlement when tempered at 400 °C and 460 °C, breaking before reaching the plastic regime. The measured Vickers hardness (HV), yield strength (YS), ultimate tensile strength (UTS) and uniform elongation (Uel) values are shown in Tables 2–6 for the LC, LM, HS, HCHS and HA steels, respectively. The hardness measurements were performed in most of the tempering conditions for all steels, except for the LM steel due to the unavailability of material. As the LC steel presented tempering embrittlement when tempered at 400 °C and 460 °C, it was not possible to determine the YS, UTS and Uel in these conditions. Finally, the as-quenched HCHS sample broke in a brittle manner and it was also impossible to obtain its UTS and Uel. For the HS steel, the evolution of the yield strength and UTS with tempering temperature and time is illustrated in Fig. 3. In Fig. 3(a), the samples were tempered for 5 min therefore all the strength variation is caused by the temperature effect. While the UTS decrease in all temperature range, the yield strength stay constant up to 400 °C and then start to decrease. On the other hand, in Fig. 3(b), the samples were tempered at 460 °C and only the effect of the time is considered. Both yield strength and UTS decrease similarly since low values of tempering times. Additionally, it is possible to observe that the effect of temperature on the strength of tempered martensite is considerably greater than the effect of time. Similar evolutions of the UTS and YS with tempering temperature and time were obtained for the other steels. The only difference is that the yield strength start to decrease at lower temperature for the LM and HA steels (approximately 300 °C instead of 400 °C for the other steels).
In the present work, five low-carbon martensitic steels were tempered at a wide range of temperatures and times and the evolutions of mechanical behavior and microstructure were characterized. From these results, two composite models were proposed and their accuracy was compared. 2. Materials and methods Five steels with different combinations of C, Mn, Si and Al were produced using vacuum induction melting. Their compositions are given in Table 1. The notation employed in Table 1 is LC for the low C (0.1%), LM for the low Mn (1.8%), HS for the high Si (1.4%), HCHS for the high C and high Si (0.25%C and 1.6%Si) and HA for the high Al (1.1%) steel. The ingots were then hot rolled to approximately 4 mm thickness and coiled at 550 °C. Hot rolled sheets were cold rolled to about 1 mm thickness. Dilatometer tests and thermodynamic simulations were performed to determine the austenitization conditions. Further, samples for all the steels, except the HA one, were austenitized at 870 °C (the HA steel was austenitized at 950 °C) for 100 s and then water quenched. Thereafter, tempering treatments were conducted in oil baths for tempering at 150 °C and in salt pots for all other temperatures (from 200 °C to 500 °C). Holding time was varied from 10 s to 3 h. At the end of holding the samples were air cooled. Three tensile tests at constant strain rate of 0.007 s−1 were performed for each tempering condition of different steels. The samples gauge length was 50 mm and the width was 12.5 mm, according to the Table 1 Measured chemical composition of alloys, in wt%, balance Fe. Steel
C
Mn
Si
Al
Cr
Nb
LC LM HS HCHS HA
0.10 0.21 0.21 0.25 0.26
2.46 1.76 2.22 2.43 2.46
0.30 0.27 1.44 1.57 0.00
0.01 0.05 0.04 0.02 1.13
0.00 0.00 0.21 0.00 0.00
0.04 0.00 0.00 0.00 0.00
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Fig. 2. Experimental true stress-true strain curves of some tempering conditions performed with the LC, HCHS, HA and LM steels.
Table 2 Measured mechanical properties of the LC steel: Vickers hardness (HV), yield strength (YS0.2), ultimate tensile strength (UTS), uniform elongation (Uel), respectively.
Table 3 Measured mechanical properties of the LM steel: yield strength (YS0.2), ultimate tensile strength (UTS), uniform elongation (Uel), respectively.
LC steel
HV
YS0.2[MPa]
UTS [MPa]
Uel[%]
LM steel
YS0.2[MPa]
UTS [MPa]
Uel[%]
As-quenched 150 °C 300 s 150 °C 3600 s 150 °C 10,800 s 200 °C 300 s 230 °C 300 s 300 °C 300 s 300 °C 3600 s 300 °C 10,800 s 400 °C 10 s 400 °C 50 s 400 °C 100 s 400 °C 300 s 400 °C 3600 s 460 °C 10 s 460 °C 50 s
430 429 389 441 403 394 361 396 390 367 – – 356 343 – 330
1175 1153 1176 1187 1197 1158 1178 1192 1171 1149 1142 – – – – –
1350 1364 1333 1345 1338 1290 1272 1264 1242 1179 1167 – – – – –
2.3 3.8 2.7 3.4 3.8 3.5 3.2 3.0 3.4 2.5 2.4 – – – – –
150 °C 150 °C 150 °C 230 °C 300 °C 300 °C 300 °C 400 °C 400 °C 460 °C 460 °C
1454 1428 1427 1357 1285 1274 1262 1248 1224 1140 1074
1748 1710 1692 1631 1461 1415 1397 1330 1278 1177 1114
3.7 3.2 3.4 4.0 3.2 2.7 3.1 3.0 3.2 3.0 3.8
300 s 3600 s 10,800 s 300 s 300 s 3600 s 10,800 s 60 s 300 s 60 s 300 s
characteristic hardness by Hollomon and Jaffe). Therefore, the linear parts of the curves can be represented by the following equation:
HV = HVc −0.039T (13+logt ) 4. Discussion
(3)
The values found for HVc in this work are 770, 830, 870 and 810 for the LC, HS, HCHS and HA steels, respectively. Interestingly, the decrease of HV start approximately at the same tempering parameter. Similarly to the hardness, all mechanical properties (yield strength, ultimate tensile strength and uniform elongation) appear to vary linearly with the tempering parameter as seen in Fig. 4(b), (c) and (d). The yield strength remains constant up to different values of tempering parameter for each steel (9760 for the LC, 10,416 for the HS and the HCHS, and 7785 for the LM and the HA). Two tendencies can be observed: the higher the silicon content or the smaller the carbon content, the higher is the critical tempering parameter (13 + logt ) at which the decrease of the yield strength begins. It can be also observed that the trend lines for the two silicon steels (HS and HCHS) have
4.1. Characterization of mechanical properties The mechanical properties displayed at Tables 2–6 are plotted in Fig. 4 against the tempering parameter [(13+logt)T] proposed by Hollomon and Jaffe [14], where the temperature is in Kelvin and the time is in seconds. The error bars in Fig. 4(a) represent 95% confidence intervals. From Fig. 4(a) it can be seen that the values of hardness stay constant up to the tempering parameter of around 8000 and then start to decrease linearly. The slope is the same for all steels and the only value that varies with the composition is the HVc constant (named 40
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different slopes from the others. The influence of silicon on retarding the softening of tempered martensite has already been reported in literature. Silicon is rejected from cementite at early stages of tempering. As the partitioning of this substitutional element is time-consuming, silicon slows the growth of cementite [27,28]. Fig. 5 shows the carbides of tempered samples of the HS and LC steels and confirms indirectly the retarding effect of silicon on cementite growth. The precipitates of the higher silicon HS steel are finer than the ones of the LC, although the HS was tempered for a longer time (1 h instead of 5 min at the same temperature 460 °C). The relations between yield strength and tempering parameter can be represented as follows:
Table 4 Measured mechanical properties of the HS steel: Vickers hardness (HV), yield strength (YS0.2), ultimate tensile strength (UTS), uniform elongation (Uel), respectively. HS steel
HV
YS0.2[MPa]
UTS [MPa]
Uel[%]
As-quenched 150 °C 300 s 150 °C 3600 s 150 °C 10,800 s 200 °C 300 s 230 °C 300 s 300 °C 300 s 300 °C 3600 s 300 °C 10,800 s 400 °C 10 s 400 °C 30 s 400 °C 50 s 400 °C 100 s 400 °C 300 s 400 °C 3600 s 460 °C 10 s 460 °C 30 s 460 °C 50 s 460 °C 100 s 460 °C 300 s 460 °C 1200 s 460 °C 2700 s 460 °C 3600 s 500 °C 10 s 500 °C 100 s 500 °C 300 s 500 °C 1200 s 500 °C 2700 s 500 °C 3600 s
536 532 539 535 520 568 497 480 493 463 – – – – 451 423 – – – 398 383 – 373 – 358 364 – – 334
1388 1393 1388 1396 1394 1368 1385 1402 1396 1421 1414 1415 1392 1367 1296 1302 1262 1285 1265 1228 1168 1129 1121 1180 1146 1092 1041 997 992
1851 1776 1734 1698 1721 1687 1630 1624 1607 1596 1588 1577 1563 1529 1439 1431 1410 1396 1363 1342 1256 1219 1224 1267 1223 1193 1149 1107 1112
4.2 3.9 3.9 3.5 4.3 4.5 4.0 3.8 4.0 3.5 3.6 3.4 3.5 3.3 5.3 4.0 3.8 4.2 3.3 5.2 5.6 5.8 5.9 4.9 4.8 6.4 6.8 6.4 7.2
HV
YS0.2[MPa]
UTS [MPa]
Uel[%]
As-quenched 150 °C 300 s 150 °C 3600 s 150 °C 10,800 s 230 °C 300 s 300 °C 300 s 300 °C 3600 s 300 °C 10,800 s 400 °C 10 s 400 °C 300 s 400 °C 1200 s 400 °C 3600 s 460 °C 10 s 460 °C 300 s 460 °C 1200 s 460 °C 3600 s
570 – 567 – 543 513 – – – – – 456 – – – 394
1490 1600 1466 1593 1496 1531 1519 1486 1467 1356 1419 1373 1418 1305 1247 1189
– 1906 1851 1761 1758 1715 1710 1679 1672 1600 1534 1482 1518 1401 1333 1290
– 3.9 4.0 3.6 3.5 3.1 3.1 3.2 3.4 3.3 3.2 3.5 3.5 4.2 5.4 6
HV
YS0.2[MPa]
UTS [MPa]
Uel[%]
As-quenched 150 °C 3600 s 230 °C 300 s 300 °C 300 s 300 °C 3600 s 400 °C 300 s 400 °C 3600 s 460 °C 10 s 460 °C 300 s 460 °C 3600 s
531 520 – 469 – 416 389 – – 328
1378 1420 1394 1209 1202 1160 1104 1100 1042 939
1865 1683 1565 1472 1428 1314 1228 1229 1124 1046
4.0 3.6 3.8 3.7 3.4 4.0 3.9 3.9 4.5 5.5
σy0.2% = 2820 − 0.14T (13+logt )
(5)
UTS = UTSc −0.12T (13+logt )
(6)
Uel = −11.85 + 0.00147T (13+logt )
(7)
In the Eq. (6), UTSc is a constant equal to 2700 for the silicon containing steels (HS and HCHS) and 2532 for the other steels (LC, HA and LM). 4.2. Modeling 4.2.1. Simplified CCA model As a starting point it was proposed to use the simplified Continuous Composite Approach (CCA) [29,30], which provides the following equations for the strain hardening of as-quenched martensite:
d∑_ = Y (1−F (σ )) dE
(8) n
F (σ ) = 1 − exp (− ⎛ ⎝ ⎜
σ − σmin ⎞ ) σ0 ⎠ ⎟
(9)
where Σ is the macroscopic stress, E is the macroscopic strain, Y is the Young's modulus. F(σ) is a cumulated fraction of a stress spectrum calculated considering σmin and n to be constants (equal to 450 MPa and 2.5, respectively) and σ0 dependent on the initial carbon (C) and manganese (Mn) contents as shown in the Eq. (10) and (11).
Table 6 Measured mechanical properties of the HA steel: Vickers hardness (HV), yield strength (YS0.2), ultimate tensile strength (UTS), uniform elongation (Uel), respectively. HA steel
(4)
where the Eq. (4) is valid for the steels with negligible amount of silicon (LC, LM and HA), while the Eq. (5) is valid only for the silicon containing steels (HS and HCHS – around 1.5%). From Fig. 4(b), it can be also seen that the yield strengths of all steels tend to the same value of 700 MPa at tempering parameter approximately equal to 15,000. This can be explained by the fact that at very high tempering levels the softest constituent of microstructure, recovered martensite (ferrite), should be more or less the same. Finally, the values of UTS and the uniform elongation can be also described by the following linear equation as a function of the tempering parameter:
Table 5 Measured mechanical properties of the HCHS steel: Vickers hardness (HV), yield strength (YS0.2), ultimate tensile strength (UTS), uniform elongation (Uel), respectively. HCHS steel
σy0.2% = 2100 − 0.093T (13+logt )
σ0 = 130 + 1997*Ceq
Ceq = C *(1+
Mn ) 3, 5
(10) (11)
Firstly, the simplified CCA model – Eqs. (8)–(11) – was tested for the as-quenched LC, HS, HCHS and HA steels. Fig. 6 presents the experimental tensile curves from the tests performed in the present work and the ones calculated by the CCA model. It is possible to observe that the model predicts well the curves for the silicon containing steels (HS and HCHS). As the CCA model was built for carbon-manganese steels, it was not expected that it would work very well for these silicon containing steels. On the contrary, it was expected that the model would provide curves softer than the experimental ones due to the solid solution strengthening effect of silicon. On the other hand, the model predicts a 41
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Fig. 3. Evolution of the yield strength and UTS with (a) temperature and (b) time for the HS steel.
tempering. In fact, the percentage of carbon segregated to the defects is also included in the Css parameter, i.e. the Css is the percentage of carbon that is not precipitated as carbides. This consideration is supported by the Hutchinson et al. [18] conclusion that segregated carbon atoms have the same influence on strengthening as carbon atoms at interstitials sites, or possibly greater. The Css is expected to reduce during tempering due to carbides formation. Therefore, the Eq. (11) of the CCA model is modified in the following manner:
softer martensite for the LC steel. This difference is probably due to the Niobium strengthening effect that is not taken into account in the model. Finally, the worst agreement between the experimental curve and the one predicted by the model is for the HA steel, in which the model predicts a much higher strain hardening rate. This disagreement indicates the possibility that this alloy has suffered auto-tempering. 4.2.2. One fitting parameter model In the simplified CCA model martensite is modeled only as a function of the equivalent carbon content (Ceq) which takes into account the concomitant effect of initial carbon and manganese contents. In the present work the initial carbon content value of Eq. (11) is substituted by the carbon content in solid solution (Css) in order to take into account the precipitation of carbon as carbides that occurs during
Ceq = Css *(1+
Mn ) 3.5
(12)
The Ceq calculated by the Eq. (12) is then introduced in the Eq. (10). Thereafter, the cumulated fraction of the stress spectrum and the
Fig. 4. Measured mechanical properties (a) Vickers hardness (HV), (b) yield strength (YS0.2), (c) ultimate tensile strength (UTS) and (d) uniform elongation (Uel) as a function of the tempering parameter.
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Fig. 5. SEM images showing carbides at (a) LC steel tempered at 460 °C for 300 s and (b) HS steel tempered at 460 °C for 3600 s.
Firstly, the normalized Css values used to fit the tensile curves with the CCA model adjusted for tempered martensite (using Eq. (12)) were plotted against the tempering parameter as shown in Fig. 7. The initial carbon content was adjusted for the LC and HA steels in order to fit their as-quenched tensile curves well with the CCA model (see Fig. 6). Therefore, the values of initial carbon used to plot Fig. 7 were equal to 0.11% for the LC steel and 0.21% for the HA one in order to disregard the effects of auto-tempering and Niobium strengthening in the model as it was discussed in Section 4.2.1. The nominal carbon content was used for the others steels. The increase of the normalized Css term with the tempering parameter can be described by the following linear equation:
strain hardening is calculated by the Eq. (8) and (9) of the CCA model. The reduction of the Css parameter with tempering promotes a decrease in the σ0 values (Eq. (10)). As σ0 controls the width of the stress spectrum, the reduction of its value makes the spectrum narrower. This indicates that the martensite loses its mechanical heterogeneity (main characteristic of the composite model) with tempering. This observation is in agreement with the Badinier et al. work [16]. From a microstructural point of view, this increase of mechanical homogeneity during tempering can be explained by the diffusion of carbon, the relief of internal stresses and the annihilation of the cell structure (and other low angle boundaries). The next step to actually build a model for tempered martensite was to relate the variation of Css to known parameters, such as the steel composition and the tempering temperatures and times.
Fig. 6. Experimental and CCA modeled tensile curves for as-quenched HS, HCHS, LC and HA steels.
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Fig. 7. Normalized fit Css values for the LC, HS, HCHS, HA and LM steels as a function of tempering parameters.
Cinitial − Css = 0.000103T (13+logt )−0.655 Cinitial
Fig. 9. Normalized fit Css values as a function of tempering parameters, considering different constant values of σmin for each steel: 650 MPa for the HA, 750 MPa for the LC, 850 MPa for the HS and HCHS, and 910 MPa for the LM steel.
So far the σmin value in equation (9) was considered constant and equal to 450 MPa. In the present part the influence of the composition was considered varying the σmin. Changing the values of σmin for each steel, it is possible to plot the fit values of normalized Css for low tempering in one straight line for all steel compositions as shown in Fig. 9. In the CCA model, changes in σmin shift the stress spectrum and modify the values of stress at which the phases of the composite start to plasticize. For example, a decrease in σmin shifts the spectrum to lower values of stress. Therefore, for the same tensile curve, a decrease of σmin increases the Css values necessary to model well the curve (and decreases the normalized Css term). The changes of the σmin with composition made in Fig. 9 only shifted the normalized Css values down or up in order to group all of them at the same level for low tempering conditions. According to this approach, the steels with high σmin values are the ones that present slower kinetics of precipitation during low tempering and the steels with lower σmin values are the ones that present faster kinetics of precipitation during low tempering. Therefore, for low tempering levels the studied steels can be classified according to the kinetics of precipitation in the following order from the slowest to the fastest: LM, HS/HCHS, LC and HA. From this order, it is possible to infer that silicon retards the precipitation. The carbon seems to accelerate the precipitation and the effect of aluminum in retarding the precipitation predicted in the literature [31] was not observed. These tendencies have covered the kinetics of all steels except the LM. As this steel has lower manganese content, it is possible that manganese accelerates the precipitation at low levels of tempering. Leslie and Rauch
(13)
Eq. (13) provided a way to model the mechanical behavior of tempered martensite regardless the content of alloying elements (except the influence of Al and Nb on the mechanical behavior of the asquenched martensite). However, from the mechanical testing and carbides characterization, it is possible to infer that the mechanical behavior is actually dependent on the alloy elements, especially silicon. Fig. 8(a) shows some of the tempered tensile curves of Fig. 1 with their respective modeled curves predicted by the one fitting parameter model described in this section. The model presented satisfactory results for most of the curves. As well, experimental strain-hardening rate evolutions and those predicted by the one fitting parameter model for the same tensile curves are shown in Fig. 8(b). As it can be observed a good agreement between model and experimental results was obtained for short tempering at low temperature. On the other hand, this agreement is less good for the high temperature conditions because experimentally almost no strain-hardening is measured during plastic deformation and the model still proposes a certain strain hardening. Nevertheless, the strain-hardening prediction is considered to be acceptable. Further improvements to this one fitting parameter model are suggested in the next section. 4.2.3. Two fitting parameter model Another approach considers that the data in Fig. 7 can be grouped into two ranges: low tempering - where the slope of precipitation is smaller - and high tempering - where the precipitation and the consequent depletion of Css occurs much faster.
Fig. 8. Comparison of experimental and one fitting model results for some of the tempered HS steel samples: a) tensile curves and b) strain-hardening evolution.
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[31] expected that the manganese lowers the cementite initiation temperature due to its high solubility in cementite, which could be the cause of this apparent acceleration of precipitation at low tempering. On the other hand, Grange et al. [13] have observed that manganese retards the softening for tempering temperatures above 300 °C, however no information about its effect for lower tempering temperatures is mentioned. Taking into account that there was only one steel with lower manganese, it is not possible to make a clear conclusion about the effect of Mn; more detailed study should be performed. The nominal carbon content was used for all steels to plot Fig. 9. In this case, the non-agreement of the as-quenched LC and HA tensile curves with the ones predicted by the simplified CCA model was taken into account in the σmin parameter, since this deviation is probably related to the composition. The equation of the line that gives the best description of the low tempering Css variation is:
Cinitial − Css = 0.000048T (13+logt )+0.1275 Cinitial
Fig. 11. Normalized fit Css values considering σmin variation with steel composition and level of tempering.
(14) fitting parameter model provided better results than the one fitting parameter model. Fig. 12(b) presents strain-hardening rates evolutions in the same way as Fig. 8(b). The evolutions predicted by the two models are very similar. Both show slight deviation with the experimental curves mainly due to the occurrence of plateaus (absence of strain-hardening) for the high temperature tempering treatments.
In Eq. (15) the σmin varies according to the chemical composition as follows: 650 MPa for the HA, 750 MPa for the LC, 850 MPa for the HS and HCHS and 910 MPa for the LM steel. The transition of low tempering to high tempering occurs in a close range of tempering parameters in which the yield strength starts to decrease (Fig. 4b). This characteristic tempering parameter when the yield strength starts decreasing is considered as the low tempering limit and slightly varies according to the steel compositions. From the data of this work it was determined to be 7785 for the HA and LM, 9760 for the LC and 10416 for the HS and HCHS steels. At high tempering conditions, the strength of the steels decreases considerably. A decrease in σmin with tempering and a consequent shift of the stress spectrum to low values was necessary to take into account this effect. This variation of σmin with tempering temperatures and times is presented in Fig. 10 and allowed to fit all normalized Css points in one line represented by Eq. (15), as shown in Fig. 11. Although there are some plateaus in the variation of σmin, the decrease of σmin is approximately linear with the same slope for all steels:
σmin = σmin c −0.1T (13+logt )
4.2.4. Performance of the models Lastly, a comparison of the three models: the simplified CCA model for as-quenched martensite, the first model for tempered martensite (one fitting parameter with σmin constant and equal to 450 MPa) and the second model for tempered martensite (two fitting parameters: σmin variation with composition and tempering condition) is presented in Fig. 13. The graphs show that the yield strength and UTS predicted by the three models are in agreement with the experimental values. As expected, the second model for tempered martensite (with σmin variation) provides better results. However, its use requires the knowledge of the values of three constants: σmin for low tempering levels, the tempering parameter that defines the tempering regime (low or high tempering) and the σminc. These constants are related to the steel composition and are strongly dependent on the silicon content. The statistical performance of the models for tempered martensite is given in Table 7 with the values of the accuracy, bias, correlation and efficacy. These values confirm the previous observation that the second model (two fitting parameters) provides better results for the yield strength and UTS.
(15)
where σminc is a constant equal to 1520 MPa for the HA, 1650 MPa for the LC and LM, and 1800 MPa for the HS and HCHS steels. As σmin variation with the tempering parameter has approximately the same slope for all steels, it seems that no further adjustment of the kinetics of precipitation with the steel composition is required for high tempering temperatures. Similarly to Fig. 8(a), Fig. 12(a) shows some experimental tensile curves of tempered HS samples and the curves predicted by the two fitting parameter model described in this section. As expected, the two
5. Conclusions Evolution of mechanical behavior with the tempering conditions was studied on steels with different chemical compositions. Tensile properties of samples tempered in a wide range of temperatures and times were measured. Linear relationships between mechanical properties and tempering parameters were established. These relationships showed that silicon retards the decrease of strength due to tempering. CCA model for as-quenched martensite was modified to predict the mechanical behavior of tempered martensite. A carbon in solid solution term (Css) was included in the model and a linear relationship was found between this term and the tempering parameter. The performance of such model was considered to be satisfactory. Further improvements were as well proposed in order to include the influence of composition on the mechanical behavior of tempered martensite. However, this model is not straightforward and necessitates two additional steps: 1) Calculation of the tempering parameter, which is used to decide the tempering condition: low tempering (σmin constant) or high
Fig. 10. Variation of σmin with the tempering parameter for all steels.
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Fig. 12. Comparison of experimental and two fitting model results for some of the tempered HS steel samples: a) tensile curves and b) strain-hardening evolution.
Fig. 13. Comparison of the CCA model, the first model for tempered martensite (σmin constant and equals to 450 MPa) and the second model for tempered martensite (σmin variation with composition and tempering parameter) according to yield strength (YS) and ultimate tensile strength (UTS).
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[7] C.M. Wayman, The growth of martensite since E.C. Bain (1924) – some milestones, Mater. Sci. Forum 56–58 (1990) 1–32. [8] G. Krauss, Martensite in steel: strength and structure, Mater. Sci. Eng. A 40–57 (1999) 273–275. [9] G.R. Speich, W.C. Leslie, Tempering of steel, Metall. Trans. 3 (1972) 1043–1054. [10] G.R. Speich, Tempering of low-carbon martensite, Trans. Metall. Soc. AIME 245 (1969) 2553–2563. [11] R.N. Caron, G. Krauss, The tempering of Fe-C lath martensite, Metall. Trans. 3 (1972) 2381–2389. [12] T. Swarr, G. Krauss, The effect of structure in the deformation of as-quenched and tempered martenside in an Fe-0.2Pct C alloy, Metall. Trans. A 7A (1976) 41–48. [13] R.A. Grange, C.R. Hribal, L.F. Porter, Hardness of tempered martensite in carbon and low-alloy steels, Metall. Trans. A 8A (1977) 1775–1785. [14] J.H. Hollomon, L.D. Jaffe, Time-Temperature Relations in Tempering Steel, Trans. Metall. Soc. AIME, New York Meeting, 1945, pp. 223–249. [15] S. Morito, H. Yoshida, T. Maki, X. Huang, Effect of block size on the strength of lath martensite in low carbon steels, Mater. Sci. Eng. A 438–440 (2006) 237–240. [16] G. Badinier, C.W. Sinclair, X. Sauvage, X. Wang, V. Bylik, M. Gouné, F. Danoix, Microstructural heterogeneity and its relationship to the strength of martensite, Mater. Sci. Eng. A 638 (2015) 329–339. [17] T. Furuhara, K. Kobayashi, T. Maki, Control of cementite precipitation in lath martensite by rapid heating and tempering, ISIJ Int. 44 (2004) 1937–1944. [18] B. Hutchinson, J. Hagström, O. Karlsson, D. Lindell, M. Tornberg, F. Lindberg, M. Thuvander, Microstructures and hardness of as-quenched martensites (0.10.5%C), Acta Mater. 59 (2011) 5845–5858. [19] Z.M. Shi, W. Gong, Y. Tomota, S. Harjo, J. Li, B. Chi, J. Pu, Study of tempering behaviour of lath martensite using in situ neutron diffraction, Mater. Charact. 107 (2015) 29–32. [20] C. Wang, H. Qiu, Y. Kimura, T. Inoue, Morphology, crystallography, and crack paths of tempered lath martensite in a medium-carbon low-alloy steel, Mater. Sci. Eng. A 669 (2016) 48–57. [21] S.C. Kennett, G. Krauss, K.O. Findley, Prior austenite grain size and tempering effects on the dislocation of low-C Nb-Ti microalloyed lath martensite, Scr. Mater. 107 (2015) 123–126. [22] G. Masing, Zur Heyn’schen Theorie der Verfestigung der Metalle durch verborgen elastische Spannungen, Wissenschaftliche Veröffentlichungen aus dem SiemensKonzern 3, 1923, pp. 231–239. [23] O.B. Pedersen, L.M. Brown, W.M. Stobbs, The Bauschinger effect in copper, Acta Metall. 29 (1981) 1843–1850. [24] H. Mughrabi, Dislocation wall and cell structures and long-range internal stresses in deformed metal crystals, Acta Metall. 31 (1983) 1367–1379. [25] J. Polák, M. Klesnil, The hysteresis loop 1. A statistical theory, Fatigue Eng. Mater. Struct. 5 (1982) 19–32. [26] J. Polák, M. Klesnil, J. Helešic, The hysteresis loop 2. An analysis of the loop shape, Fatigue Eng. Mater. Struct. 5 (1982) 33–44. [27] G. Miyamoto, J.C. Oh, K. Hono, T. Furuhara, T. Maki, Effect of partitioning of Mn and Si on the growth kinetics of cementite in tempered Fe-0.6 mass%C martensite, Acta Mater. 55 (2007) 5027–5038. [28] S. Ghosh, Rate-controlling parameters in the coarsening kinetics of cementite in Fe0.6C steels during tempering, Scr. Mater. 63 (2010) 273–276. [29] S. Allain, O. Bouaziz, M. Takahashi, Toward a new interpretation of the mechanical behaviour of as-quenched low alloyed martensitic steels, ISIJ Int. 52 (n.4) (2012) 717–722. [30] A. Arlazarov, O. Bouaziz, A. Hazotte, M. Gouné, S. Allain, Characterization and modeling of manganese effect on strength and strain hardening of martensitic carbon steels, ISIJ Int. 53 (n.6) (2013) 1076–1080. [31] W.C. Leslie, G.C. Rauch, Precipitation of carbides in low-carbon Fe-Al-C alloys, Metall. Trans. A 9A (1978) 343–349.
Table 7 Statistical performance of the models. YS0.2
Accuracy Bias Correlation Efficacy
UTS
1st model
2nd model
1st model
2nd model
78.2 159.9 0.85 0.86
65.8 43.4 0.94 1.23
101.1 −73.0 0.92 1.18
41.4 −37.0 0.99 1.07
tempering (σmin variable). 2) Secondly, if the sample is in the low tempering condition it is necessary to find the σmin that will be used. On the other hand, if the sample is in the high tempering condition, it is necessary to obtain the σminc and calculate σmin. The higher the value of low tempering limit and the σmin for low tempering, higher the value of σminc. The yield strength and the ultimate tensile strength predicted by both models are in agreement with the experimental values. However, improvements to model can be further developed to better predict these values. As equations, required in the second model, relating the constants to the composition are not available, future investigations are planned in order to include in the model the effect of alloying elements on the softening of tempered martensite. Acknowledgements The authors wish to express their gratitude to Virginie Blesse for her help in the experimental part of this work. References [1] R. Maddin, A history of martensite: some thoughts on the early hardening of iron, in: G.B. Olson, W.S. Owen (Eds.), Martensite, ASM Int., Materials Park, OH, 1992, pp. 11–19. [2] C.S. Smith, A history of martensite: early ideas on the structure of steel, in: G.B. Olson, W.S. Owen (Eds.), Martensite, ASM Int., Materials Park, OH, 1992, pp. 21–39. [3] L. Kaufman, M. Cohen, Thermodynamics and kinetics of martensitic transformations, Prog. Met. Phys. 7 (1958) 165–246. [4] C.M. Wayman, Introduction to the Crystallography of Martensite Transformations, MacMillian, New York, NY, 1964. [5] G. Krauss, A.R. Marder, The morphology of martensite in iron alloys, Metall. Trans. 2 (1971) 2343–2357. [6] A.R. Entwisle, The kinetics of martensite formation in steel, Metall. Trans. 2 (1971) 2395–2407.
47
Contents lists available at ScienceDirect
Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
Mechanical behavior of tempered martensite: Characterization and modeling L.R. Cupertino Malheirosa,1, E.A. Pachon Rodrigueza, A. Arlazarova, a
MARK
⁎
ArcelorMittal Global Research and Development, Voie Romaine BP30320, 57283 Maizières-lès-Metz Cedex, France
A R T I C L E I N F O
A B S T R A C T
Keywords: Martensite Tempering Strain hardening Modeling Mechanical behavior
Mechanical characterization of five low-carbon martensitic steels tempered at a wide range of temperatures and times was performed. Analysis of the relationships between mechanical properties, hardness and tempering conditions was completed. Microstructural observations revealed the precipitation of carbides and consequent decrease of carbon in solid solution with tempering. To describe the mechanical behavior of tempered martensite, a simplified Continuous Composite model, reported previously, was adapted. A good agreement between the model and the experimental stress-strain curves was achieved with only one fitting parameter. Further directions for improvement of the model were also proposed.
1. Introduction
The precipitation of rod-shaped cementite starts at 250 °C and it tends to be replaced by spheroidal particles at 400 °C. The recovery of the very deformed martensitic structure occurs more significantly at temperatures above 400 °C. Finally, recrystallization and grain growth are the last stages of tempering. Caron and Krauss [11] also studied the changes of the lath martensite when tempering at high temperatures. They noticed that the elongated packet-lath morphology of martensite is maintained up to long times at high tempering temperatures. In addition, they observed the laths coarsening using electron microscopy. In another work, Swarr and Krauss [12] observed the presence of discshaped and spherical carbides with 10 nm in diameter within the lattices, and larger carbides – 20 nm in thickness and 100 nm in length - in the lath boundaries of the tempered steel. They also observed the transformations in the substructure of the as-quenched and tempered martensitic steels before and after being deformed by tensile testing. While the as-quenched sample has a cellular structure that develops during deformation, the tempered sample has a uniform distribution of dislocations that practically does not change during deformation. Tempering also impacts hardness. Grange et al. [13] observed that the hardness of as-quenched martensite depends only on the carbon content. However, the hardness of tempered martensite substantially increases with alloying elements additions. Hollomon and Jaffe [14] modeled the hardness evolution with tempering temperatures and times. They found the following relationship between the tempering conditions and the hardness:
The use of hardened steels date from thousands of years ago [1]. However, characterization and understanding of their microstructures started only after the availability of analytical and testing techniques in the end of the nineteenth century. The name martensite was given to honor the important contribution of Adolf Martens [2] in the metallographic observations of the hard phase found in the quenched steels. Now, this name is used to designate any phase in ferrous and nonferrous systems formed by diffusionless and solid-state shear transformation. In steels, martensite is a very complex phase that was studied in the twentieth century by many researchers [3–8]. Its structure varies from lath-like to plate-like, its substructure from dislocations to twins and its crystallography from BCC to BCT when the carbon content increases. Tempering process is the reheating of a martensitic steel to a certain temperature, accompanied with the holding for a certain time, in order to improve the toughness and strength-ductility balance of the martensitic product. Tempered martensite has received increasing attention since it is one of the phases present in most of advanced high strength steels (AHSS). Speich and Leslie [9,10] gave an account of the different microstructural changes that occur during tempering. Firstly, at tempering temperatures between 100 and 200 °C, the carbon segregation to the defects, started during quenching, continues to occur. Therefore, there is a depletion of carbon in the solid solution inhibiting the ε-carbide precipitation in low carbon steels. Secondly, at temperatures between 200 and 300 °C, the decomposition of retained austenite usually begins.
⁎
1
R c = Hc −0.00457T (13+logt )
Corresponding author. E-mail address: [email protected] (A. Arlazarov). Present address: Department of Metallurgical and Materials Engineering, Universidade Federal de Minas Gerais, Belo Horizonte, MG 31270-901, Brazil.
http://dx.doi.org/10.1016/j.msea.2017.08.089 Received 15 May 2017; Received in revised form 10 August 2017; Accepted 25 August 2017 Available online 26 August 2017 0921-5093/ © 2017 Elsevier B.V. All rights reserved.
(1)
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where Rc is the Rockwell C hardness, Hc is a characteristic hardness, T is the tempering temperature in Kelvin and t is the tempering time in seconds. Despite all the work performed on tempered martensite [15–21], the microstructural evolution during tempering at a fine scale and the effects of different microstructural changes on the strain-hardening of tempered martensite are still unclear. In this context, the purpose of this work is to propose a composite model for the prediction of the stress-strain curves of tempered martensitic steels. Composite models started to be applied in the 20th century. They are based on the Masing's model of polycrystal plasticity [22], in which the individual grains yield at different external stress values. Pedersen et al. [23] proposed a composite model to explain the Bauschinger effect of workhardened copper. In this model the dislocations tangles act as hard but deformable particles in a soft phase, practically free of dislocations. Mughrabi [24] proposed a composite model for the plasticity of crystals with cell structures. In his model, the cells walls are considered as hard phases separated by soft regions with low dislocation density. Polák et al. [25,26] built a more sophisticated composite model. Instead of considering just two phases as in the models of Pederson and Mughrabi, i.e., one phase with high and another with low dislocation density, this model deals with a continuous distribution of volumes with different internal critical flow stresses (σic). This distribution was characterized by a probability density function that can be written as:
∫0
∞
Fig. 1. Experimental true stress-true strain curves of some tempering conditions performed with the HS steel.
ISO procedure. The presence, size and morphology of cementite was observed with SEM after etching with Picral reagent (4 g of picric acid in 100 ml of ethanol). Finally, conventional Vickers micro-hardness measurements were performed at 13 locations of the polished surfaces using the load of 1 kg and holding time of 10 s.
3. Results
f (σic ) dσic = 1
(2) Fig. 1 shows some of the experimental tensile curves obtained in the present study for the HS steel. These curves illustrate the mechanical behavior evolution with tempering and the expected softening of the steel. Similarly, Fig. 2 presents some of the experimental tensile curves for the other steels (LC, HCHS, HA and LM). The softening behavior observed in Fig. 1 also occurs for these other steels. However, the LC steel presented some sort of temper embrittlement when tempered at 400 °C and 460 °C, breaking before reaching the plastic regime. The measured Vickers hardness (HV), yield strength (YS), ultimate tensile strength (UTS) and uniform elongation (Uel) values are shown in Tables 2–6 for the LC, LM, HS, HCHS and HA steels, respectively. The hardness measurements were performed in most of the tempering conditions for all steels, except for the LM steel due to the unavailability of material. As the LC steel presented tempering embrittlement when tempered at 400 °C and 460 °C, it was not possible to determine the YS, UTS and Uel in these conditions. Finally, the as-quenched HCHS sample broke in a brittle manner and it was also impossible to obtain its UTS and Uel. For the HS steel, the evolution of the yield strength and UTS with tempering temperature and time is illustrated in Fig. 3. In Fig. 3(a), the samples were tempered for 5 min therefore all the strength variation is caused by the temperature effect. While the UTS decrease in all temperature range, the yield strength stay constant up to 400 °C and then start to decrease. On the other hand, in Fig. 3(b), the samples were tempered at 460 °C and only the effect of the time is considered. Both yield strength and UTS decrease similarly since low values of tempering times. Additionally, it is possible to observe that the effect of temperature on the strength of tempered martensite is considerably greater than the effect of time. Similar evolutions of the UTS and YS with tempering temperature and time were obtained for the other steels. The only difference is that the yield strength start to decrease at lower temperature for the LM and HA steels (approximately 300 °C instead of 400 °C for the other steels).
In the present work, five low-carbon martensitic steels were tempered at a wide range of temperatures and times and the evolutions of mechanical behavior and microstructure were characterized. From these results, two composite models were proposed and their accuracy was compared. 2. Materials and methods Five steels with different combinations of C, Mn, Si and Al were produced using vacuum induction melting. Their compositions are given in Table 1. The notation employed in Table 1 is LC for the low C (0.1%), LM for the low Mn (1.8%), HS for the high Si (1.4%), HCHS for the high C and high Si (0.25%C and 1.6%Si) and HA for the high Al (1.1%) steel. The ingots were then hot rolled to approximately 4 mm thickness and coiled at 550 °C. Hot rolled sheets were cold rolled to about 1 mm thickness. Dilatometer tests and thermodynamic simulations were performed to determine the austenitization conditions. Further, samples for all the steels, except the HA one, were austenitized at 870 °C (the HA steel was austenitized at 950 °C) for 100 s and then water quenched. Thereafter, tempering treatments were conducted in oil baths for tempering at 150 °C and in salt pots for all other temperatures (from 200 °C to 500 °C). Holding time was varied from 10 s to 3 h. At the end of holding the samples were air cooled. Three tensile tests at constant strain rate of 0.007 s−1 were performed for each tempering condition of different steels. The samples gauge length was 50 mm and the width was 12.5 mm, according to the Table 1 Measured chemical composition of alloys, in wt%, balance Fe. Steel
C
Mn
Si
Al
Cr
Nb
LC LM HS HCHS HA
0.10 0.21 0.21 0.25 0.26
2.46 1.76 2.22 2.43 2.46
0.30 0.27 1.44 1.57 0.00
0.01 0.05 0.04 0.02 1.13
0.00 0.00 0.21 0.00 0.00
0.04 0.00 0.00 0.00 0.00
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Fig. 2. Experimental true stress-true strain curves of some tempering conditions performed with the LC, HCHS, HA and LM steels.
Table 2 Measured mechanical properties of the LC steel: Vickers hardness (HV), yield strength (YS0.2), ultimate tensile strength (UTS), uniform elongation (Uel), respectively.
Table 3 Measured mechanical properties of the LM steel: yield strength (YS0.2), ultimate tensile strength (UTS), uniform elongation (Uel), respectively.
LC steel
HV
YS0.2[MPa]
UTS [MPa]
Uel[%]
LM steel
YS0.2[MPa]
UTS [MPa]
Uel[%]
As-quenched 150 °C 300 s 150 °C 3600 s 150 °C 10,800 s 200 °C 300 s 230 °C 300 s 300 °C 300 s 300 °C 3600 s 300 °C 10,800 s 400 °C 10 s 400 °C 50 s 400 °C 100 s 400 °C 300 s 400 °C 3600 s 460 °C 10 s 460 °C 50 s
430 429 389 441 403 394 361 396 390 367 – – 356 343 – 330
1175 1153 1176 1187 1197 1158 1178 1192 1171 1149 1142 – – – – –
1350 1364 1333 1345 1338 1290 1272 1264 1242 1179 1167 – – – – –
2.3 3.8 2.7 3.4 3.8 3.5 3.2 3.0 3.4 2.5 2.4 – – – – –
150 °C 150 °C 150 °C 230 °C 300 °C 300 °C 300 °C 400 °C 400 °C 460 °C 460 °C
1454 1428 1427 1357 1285 1274 1262 1248 1224 1140 1074
1748 1710 1692 1631 1461 1415 1397 1330 1278 1177 1114
3.7 3.2 3.4 4.0 3.2 2.7 3.1 3.0 3.2 3.0 3.8
300 s 3600 s 10,800 s 300 s 300 s 3600 s 10,800 s 60 s 300 s 60 s 300 s
characteristic hardness by Hollomon and Jaffe). Therefore, the linear parts of the curves can be represented by the following equation:
HV = HVc −0.039T (13+logt ) 4. Discussion
(3)
The values found for HVc in this work are 770, 830, 870 and 810 for the LC, HS, HCHS and HA steels, respectively. Interestingly, the decrease of HV start approximately at the same tempering parameter. Similarly to the hardness, all mechanical properties (yield strength, ultimate tensile strength and uniform elongation) appear to vary linearly with the tempering parameter as seen in Fig. 4(b), (c) and (d). The yield strength remains constant up to different values of tempering parameter for each steel (9760 for the LC, 10,416 for the HS and the HCHS, and 7785 for the LM and the HA). Two tendencies can be observed: the higher the silicon content or the smaller the carbon content, the higher is the critical tempering parameter (13 + logt ) at which the decrease of the yield strength begins. It can be also observed that the trend lines for the two silicon steels (HS and HCHS) have
4.1. Characterization of mechanical properties The mechanical properties displayed at Tables 2–6 are plotted in Fig. 4 against the tempering parameter [(13+logt)T] proposed by Hollomon and Jaffe [14], where the temperature is in Kelvin and the time is in seconds. The error bars in Fig. 4(a) represent 95% confidence intervals. From Fig. 4(a) it can be seen that the values of hardness stay constant up to the tempering parameter of around 8000 and then start to decrease linearly. The slope is the same for all steels and the only value that varies with the composition is the HVc constant (named 40
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different slopes from the others. The influence of silicon on retarding the softening of tempered martensite has already been reported in literature. Silicon is rejected from cementite at early stages of tempering. As the partitioning of this substitutional element is time-consuming, silicon slows the growth of cementite [27,28]. Fig. 5 shows the carbides of tempered samples of the HS and LC steels and confirms indirectly the retarding effect of silicon on cementite growth. The precipitates of the higher silicon HS steel are finer than the ones of the LC, although the HS was tempered for a longer time (1 h instead of 5 min at the same temperature 460 °C). The relations between yield strength and tempering parameter can be represented as follows:
Table 4 Measured mechanical properties of the HS steel: Vickers hardness (HV), yield strength (YS0.2), ultimate tensile strength (UTS), uniform elongation (Uel), respectively. HS steel
HV
YS0.2[MPa]
UTS [MPa]
Uel[%]
As-quenched 150 °C 300 s 150 °C 3600 s 150 °C 10,800 s 200 °C 300 s 230 °C 300 s 300 °C 300 s 300 °C 3600 s 300 °C 10,800 s 400 °C 10 s 400 °C 30 s 400 °C 50 s 400 °C 100 s 400 °C 300 s 400 °C 3600 s 460 °C 10 s 460 °C 30 s 460 °C 50 s 460 °C 100 s 460 °C 300 s 460 °C 1200 s 460 °C 2700 s 460 °C 3600 s 500 °C 10 s 500 °C 100 s 500 °C 300 s 500 °C 1200 s 500 °C 2700 s 500 °C 3600 s
536 532 539 535 520 568 497 480 493 463 – – – – 451 423 – – – 398 383 – 373 – 358 364 – – 334
1388 1393 1388 1396 1394 1368 1385 1402 1396 1421 1414 1415 1392 1367 1296 1302 1262 1285 1265 1228 1168 1129 1121 1180 1146 1092 1041 997 992
1851 1776 1734 1698 1721 1687 1630 1624 1607 1596 1588 1577 1563 1529 1439 1431 1410 1396 1363 1342 1256 1219 1224 1267 1223 1193 1149 1107 1112
4.2 3.9 3.9 3.5 4.3 4.5 4.0 3.8 4.0 3.5 3.6 3.4 3.5 3.3 5.3 4.0 3.8 4.2 3.3 5.2 5.6 5.8 5.9 4.9 4.8 6.4 6.8 6.4 7.2
HV
YS0.2[MPa]
UTS [MPa]
Uel[%]
As-quenched 150 °C 300 s 150 °C 3600 s 150 °C 10,800 s 230 °C 300 s 300 °C 300 s 300 °C 3600 s 300 °C 10,800 s 400 °C 10 s 400 °C 300 s 400 °C 1200 s 400 °C 3600 s 460 °C 10 s 460 °C 300 s 460 °C 1200 s 460 °C 3600 s
570 – 567 – 543 513 – – – – – 456 – – – 394
1490 1600 1466 1593 1496 1531 1519 1486 1467 1356 1419 1373 1418 1305 1247 1189
– 1906 1851 1761 1758 1715 1710 1679 1672 1600 1534 1482 1518 1401 1333 1290
– 3.9 4.0 3.6 3.5 3.1 3.1 3.2 3.4 3.3 3.2 3.5 3.5 4.2 5.4 6
HV
YS0.2[MPa]
UTS [MPa]
Uel[%]
As-quenched 150 °C 3600 s 230 °C 300 s 300 °C 300 s 300 °C 3600 s 400 °C 300 s 400 °C 3600 s 460 °C 10 s 460 °C 300 s 460 °C 3600 s
531 520 – 469 – 416 389 – – 328
1378 1420 1394 1209 1202 1160 1104 1100 1042 939
1865 1683 1565 1472 1428 1314 1228 1229 1124 1046
4.0 3.6 3.8 3.7 3.4 4.0 3.9 3.9 4.5 5.5
σy0.2% = 2820 − 0.14T (13+logt )
(5)
UTS = UTSc −0.12T (13+logt )
(6)
Uel = −11.85 + 0.00147T (13+logt )
(7)
In the Eq. (6), UTSc is a constant equal to 2700 for the silicon containing steels (HS and HCHS) and 2532 for the other steels (LC, HA and LM). 4.2. Modeling 4.2.1. Simplified CCA model As a starting point it was proposed to use the simplified Continuous Composite Approach (CCA) [29,30], which provides the following equations for the strain hardening of as-quenched martensite:
d∑_ = Y (1−F (σ )) dE
(8) n
F (σ ) = 1 − exp (− ⎛ ⎝ ⎜
σ − σmin ⎞ ) σ0 ⎠ ⎟
(9)
where Σ is the macroscopic stress, E is the macroscopic strain, Y is the Young's modulus. F(σ) is a cumulated fraction of a stress spectrum calculated considering σmin and n to be constants (equal to 450 MPa and 2.5, respectively) and σ0 dependent on the initial carbon (C) and manganese (Mn) contents as shown in the Eq. (10) and (11).
Table 6 Measured mechanical properties of the HA steel: Vickers hardness (HV), yield strength (YS0.2), ultimate tensile strength (UTS), uniform elongation (Uel), respectively. HA steel
(4)
where the Eq. (4) is valid for the steels with negligible amount of silicon (LC, LM and HA), while the Eq. (5) is valid only for the silicon containing steels (HS and HCHS – around 1.5%). From Fig. 4(b), it can be also seen that the yield strengths of all steels tend to the same value of 700 MPa at tempering parameter approximately equal to 15,000. This can be explained by the fact that at very high tempering levels the softest constituent of microstructure, recovered martensite (ferrite), should be more or less the same. Finally, the values of UTS and the uniform elongation can be also described by the following linear equation as a function of the tempering parameter:
Table 5 Measured mechanical properties of the HCHS steel: Vickers hardness (HV), yield strength (YS0.2), ultimate tensile strength (UTS), uniform elongation (Uel), respectively. HCHS steel
σy0.2% = 2100 − 0.093T (13+logt )
σ0 = 130 + 1997*Ceq
Ceq = C *(1+
Mn ) 3, 5
(10) (11)
Firstly, the simplified CCA model – Eqs. (8)–(11) – was tested for the as-quenched LC, HS, HCHS and HA steels. Fig. 6 presents the experimental tensile curves from the tests performed in the present work and the ones calculated by the CCA model. It is possible to observe that the model predicts well the curves for the silicon containing steels (HS and HCHS). As the CCA model was built for carbon-manganese steels, it was not expected that it would work very well for these silicon containing steels. On the contrary, it was expected that the model would provide curves softer than the experimental ones due to the solid solution strengthening effect of silicon. On the other hand, the model predicts a 41
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Fig. 3. Evolution of the yield strength and UTS with (a) temperature and (b) time for the HS steel.
tempering. In fact, the percentage of carbon segregated to the defects is also included in the Css parameter, i.e. the Css is the percentage of carbon that is not precipitated as carbides. This consideration is supported by the Hutchinson et al. [18] conclusion that segregated carbon atoms have the same influence on strengthening as carbon atoms at interstitials sites, or possibly greater. The Css is expected to reduce during tempering due to carbides formation. Therefore, the Eq. (11) of the CCA model is modified in the following manner:
softer martensite for the LC steel. This difference is probably due to the Niobium strengthening effect that is not taken into account in the model. Finally, the worst agreement between the experimental curve and the one predicted by the model is for the HA steel, in which the model predicts a much higher strain hardening rate. This disagreement indicates the possibility that this alloy has suffered auto-tempering. 4.2.2. One fitting parameter model In the simplified CCA model martensite is modeled only as a function of the equivalent carbon content (Ceq) which takes into account the concomitant effect of initial carbon and manganese contents. In the present work the initial carbon content value of Eq. (11) is substituted by the carbon content in solid solution (Css) in order to take into account the precipitation of carbon as carbides that occurs during
Ceq = Css *(1+
Mn ) 3.5
(12)
The Ceq calculated by the Eq. (12) is then introduced in the Eq. (10). Thereafter, the cumulated fraction of the stress spectrum and the
Fig. 4. Measured mechanical properties (a) Vickers hardness (HV), (b) yield strength (YS0.2), (c) ultimate tensile strength (UTS) and (d) uniform elongation (Uel) as a function of the tempering parameter.
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Fig. 5. SEM images showing carbides at (a) LC steel tempered at 460 °C for 300 s and (b) HS steel tempered at 460 °C for 3600 s.
Firstly, the normalized Css values used to fit the tensile curves with the CCA model adjusted for tempered martensite (using Eq. (12)) were plotted against the tempering parameter as shown in Fig. 7. The initial carbon content was adjusted for the LC and HA steels in order to fit their as-quenched tensile curves well with the CCA model (see Fig. 6). Therefore, the values of initial carbon used to plot Fig. 7 were equal to 0.11% for the LC steel and 0.21% for the HA one in order to disregard the effects of auto-tempering and Niobium strengthening in the model as it was discussed in Section 4.2.1. The nominal carbon content was used for the others steels. The increase of the normalized Css term with the tempering parameter can be described by the following linear equation:
strain hardening is calculated by the Eq. (8) and (9) of the CCA model. The reduction of the Css parameter with tempering promotes a decrease in the σ0 values (Eq. (10)). As σ0 controls the width of the stress spectrum, the reduction of its value makes the spectrum narrower. This indicates that the martensite loses its mechanical heterogeneity (main characteristic of the composite model) with tempering. This observation is in agreement with the Badinier et al. work [16]. From a microstructural point of view, this increase of mechanical homogeneity during tempering can be explained by the diffusion of carbon, the relief of internal stresses and the annihilation of the cell structure (and other low angle boundaries). The next step to actually build a model for tempered martensite was to relate the variation of Css to known parameters, such as the steel composition and the tempering temperatures and times.
Fig. 6. Experimental and CCA modeled tensile curves for as-quenched HS, HCHS, LC and HA steels.
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Fig. 7. Normalized fit Css values for the LC, HS, HCHS, HA and LM steels as a function of tempering parameters.
Cinitial − Css = 0.000103T (13+logt )−0.655 Cinitial
Fig. 9. Normalized fit Css values as a function of tempering parameters, considering different constant values of σmin for each steel: 650 MPa for the HA, 750 MPa for the LC, 850 MPa for the HS and HCHS, and 910 MPa for the LM steel.
So far the σmin value in equation (9) was considered constant and equal to 450 MPa. In the present part the influence of the composition was considered varying the σmin. Changing the values of σmin for each steel, it is possible to plot the fit values of normalized Css for low tempering in one straight line for all steel compositions as shown in Fig. 9. In the CCA model, changes in σmin shift the stress spectrum and modify the values of stress at which the phases of the composite start to plasticize. For example, a decrease in σmin shifts the spectrum to lower values of stress. Therefore, for the same tensile curve, a decrease of σmin increases the Css values necessary to model well the curve (and decreases the normalized Css term). The changes of the σmin with composition made in Fig. 9 only shifted the normalized Css values down or up in order to group all of them at the same level for low tempering conditions. According to this approach, the steels with high σmin values are the ones that present slower kinetics of precipitation during low tempering and the steels with lower σmin values are the ones that present faster kinetics of precipitation during low tempering. Therefore, for low tempering levels the studied steels can be classified according to the kinetics of precipitation in the following order from the slowest to the fastest: LM, HS/HCHS, LC and HA. From this order, it is possible to infer that silicon retards the precipitation. The carbon seems to accelerate the precipitation and the effect of aluminum in retarding the precipitation predicted in the literature [31] was not observed. These tendencies have covered the kinetics of all steels except the LM. As this steel has lower manganese content, it is possible that manganese accelerates the precipitation at low levels of tempering. Leslie and Rauch
(13)
Eq. (13) provided a way to model the mechanical behavior of tempered martensite regardless the content of alloying elements (except the influence of Al and Nb on the mechanical behavior of the asquenched martensite). However, from the mechanical testing and carbides characterization, it is possible to infer that the mechanical behavior is actually dependent on the alloy elements, especially silicon. Fig. 8(a) shows some of the tempered tensile curves of Fig. 1 with their respective modeled curves predicted by the one fitting parameter model described in this section. The model presented satisfactory results for most of the curves. As well, experimental strain-hardening rate evolutions and those predicted by the one fitting parameter model for the same tensile curves are shown in Fig. 8(b). As it can be observed a good agreement between model and experimental results was obtained for short tempering at low temperature. On the other hand, this agreement is less good for the high temperature conditions because experimentally almost no strain-hardening is measured during plastic deformation and the model still proposes a certain strain hardening. Nevertheless, the strain-hardening prediction is considered to be acceptable. Further improvements to this one fitting parameter model are suggested in the next section. 4.2.3. Two fitting parameter model Another approach considers that the data in Fig. 7 can be grouped into two ranges: low tempering - where the slope of precipitation is smaller - and high tempering - where the precipitation and the consequent depletion of Css occurs much faster.
Fig. 8. Comparison of experimental and one fitting model results for some of the tempered HS steel samples: a) tensile curves and b) strain-hardening evolution.
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[31] expected that the manganese lowers the cementite initiation temperature due to its high solubility in cementite, which could be the cause of this apparent acceleration of precipitation at low tempering. On the other hand, Grange et al. [13] have observed that manganese retards the softening for tempering temperatures above 300 °C, however no information about its effect for lower tempering temperatures is mentioned. Taking into account that there was only one steel with lower manganese, it is not possible to make a clear conclusion about the effect of Mn; more detailed study should be performed. The nominal carbon content was used for all steels to plot Fig. 9. In this case, the non-agreement of the as-quenched LC and HA tensile curves with the ones predicted by the simplified CCA model was taken into account in the σmin parameter, since this deviation is probably related to the composition. The equation of the line that gives the best description of the low tempering Css variation is:
Cinitial − Css = 0.000048T (13+logt )+0.1275 Cinitial
Fig. 11. Normalized fit Css values considering σmin variation with steel composition and level of tempering.
(14) fitting parameter model provided better results than the one fitting parameter model. Fig. 12(b) presents strain-hardening rates evolutions in the same way as Fig. 8(b). The evolutions predicted by the two models are very similar. Both show slight deviation with the experimental curves mainly due to the occurrence of plateaus (absence of strain-hardening) for the high temperature tempering treatments.
In Eq. (15) the σmin varies according to the chemical composition as follows: 650 MPa for the HA, 750 MPa for the LC, 850 MPa for the HS and HCHS and 910 MPa for the LM steel. The transition of low tempering to high tempering occurs in a close range of tempering parameters in which the yield strength starts to decrease (Fig. 4b). This characteristic tempering parameter when the yield strength starts decreasing is considered as the low tempering limit and slightly varies according to the steel compositions. From the data of this work it was determined to be 7785 for the HA and LM, 9760 for the LC and 10416 for the HS and HCHS steels. At high tempering conditions, the strength of the steels decreases considerably. A decrease in σmin with tempering and a consequent shift of the stress spectrum to low values was necessary to take into account this effect. This variation of σmin with tempering temperatures and times is presented in Fig. 10 and allowed to fit all normalized Css points in one line represented by Eq. (15), as shown in Fig. 11. Although there are some plateaus in the variation of σmin, the decrease of σmin is approximately linear with the same slope for all steels:
σmin = σmin c −0.1T (13+logt )
4.2.4. Performance of the models Lastly, a comparison of the three models: the simplified CCA model for as-quenched martensite, the first model for tempered martensite (one fitting parameter with σmin constant and equal to 450 MPa) and the second model for tempered martensite (two fitting parameters: σmin variation with composition and tempering condition) is presented in Fig. 13. The graphs show that the yield strength and UTS predicted by the three models are in agreement with the experimental values. As expected, the second model for tempered martensite (with σmin variation) provides better results. However, its use requires the knowledge of the values of three constants: σmin for low tempering levels, the tempering parameter that defines the tempering regime (low or high tempering) and the σminc. These constants are related to the steel composition and are strongly dependent on the silicon content. The statistical performance of the models for tempered martensite is given in Table 7 with the values of the accuracy, bias, correlation and efficacy. These values confirm the previous observation that the second model (two fitting parameters) provides better results for the yield strength and UTS.
(15)
where σminc is a constant equal to 1520 MPa for the HA, 1650 MPa for the LC and LM, and 1800 MPa for the HS and HCHS steels. As σmin variation with the tempering parameter has approximately the same slope for all steels, it seems that no further adjustment of the kinetics of precipitation with the steel composition is required for high tempering temperatures. Similarly to Fig. 8(a), Fig. 12(a) shows some experimental tensile curves of tempered HS samples and the curves predicted by the two fitting parameter model described in this section. As expected, the two
5. Conclusions Evolution of mechanical behavior with the tempering conditions was studied on steels with different chemical compositions. Tensile properties of samples tempered in a wide range of temperatures and times were measured. Linear relationships between mechanical properties and tempering parameters were established. These relationships showed that silicon retards the decrease of strength due to tempering. CCA model for as-quenched martensite was modified to predict the mechanical behavior of tempered martensite. A carbon in solid solution term (Css) was included in the model and a linear relationship was found between this term and the tempering parameter. The performance of such model was considered to be satisfactory. Further improvements were as well proposed in order to include the influence of composition on the mechanical behavior of tempered martensite. However, this model is not straightforward and necessitates two additional steps: 1) Calculation of the tempering parameter, which is used to decide the tempering condition: low tempering (σmin constant) or high
Fig. 10. Variation of σmin with the tempering parameter for all steels.
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Fig. 12. Comparison of experimental and two fitting model results for some of the tempered HS steel samples: a) tensile curves and b) strain-hardening evolution.
Fig. 13. Comparison of the CCA model, the first model for tempered martensite (σmin constant and equals to 450 MPa) and the second model for tempered martensite (σmin variation with composition and tempering parameter) according to yield strength (YS) and ultimate tensile strength (UTS).
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[7] C.M. Wayman, The growth of martensite since E.C. Bain (1924) – some milestones, Mater. Sci. Forum 56–58 (1990) 1–32. [8] G. Krauss, Martensite in steel: strength and structure, Mater. Sci. Eng. A 40–57 (1999) 273–275. [9] G.R. Speich, W.C. Leslie, Tempering of steel, Metall. Trans. 3 (1972) 1043–1054. [10] G.R. Speich, Tempering of low-carbon martensite, Trans. Metall. Soc. AIME 245 (1969) 2553–2563. [11] R.N. Caron, G. Krauss, The tempering of Fe-C lath martensite, Metall. Trans. 3 (1972) 2381–2389. [12] T. Swarr, G. Krauss, The effect of structure in the deformation of as-quenched and tempered martenside in an Fe-0.2Pct C alloy, Metall. Trans. A 7A (1976) 41–48. [13] R.A. Grange, C.R. Hribal, L.F. Porter, Hardness of tempered martensite in carbon and low-alloy steels, Metall. Trans. A 8A (1977) 1775–1785. [14] J.H. Hollomon, L.D. Jaffe, Time-Temperature Relations in Tempering Steel, Trans. Metall. Soc. AIME, New York Meeting, 1945, pp. 223–249. [15] S. Morito, H. Yoshida, T. Maki, X. Huang, Effect of block size on the strength of lath martensite in low carbon steels, Mater. Sci. Eng. A 438–440 (2006) 237–240. [16] G. Badinier, C.W. Sinclair, X. Sauvage, X. Wang, V. Bylik, M. Gouné, F. Danoix, Microstructural heterogeneity and its relationship to the strength of martensite, Mater. Sci. Eng. A 638 (2015) 329–339. [17] T. Furuhara, K. Kobayashi, T. Maki, Control of cementite precipitation in lath martensite by rapid heating and tempering, ISIJ Int. 44 (2004) 1937–1944. [18] B. Hutchinson, J. Hagström, O. Karlsson, D. Lindell, M. Tornberg, F. Lindberg, M. Thuvander, Microstructures and hardness of as-quenched martensites (0.10.5%C), Acta Mater. 59 (2011) 5845–5858. [19] Z.M. Shi, W. Gong, Y. Tomota, S. Harjo, J. Li, B. Chi, J. Pu, Study of tempering behaviour of lath martensite using in situ neutron diffraction, Mater. Charact. 107 (2015) 29–32. [20] C. Wang, H. Qiu, Y. Kimura, T. Inoue, Morphology, crystallography, and crack paths of tempered lath martensite in a medium-carbon low-alloy steel, Mater. Sci. Eng. A 669 (2016) 48–57. [21] S.C. Kennett, G. Krauss, K.O. Findley, Prior austenite grain size and tempering effects on the dislocation of low-C Nb-Ti microalloyed lath martensite, Scr. Mater. 107 (2015) 123–126. [22] G. Masing, Zur Heyn’schen Theorie der Verfestigung der Metalle durch verborgen elastische Spannungen, Wissenschaftliche Veröffentlichungen aus dem SiemensKonzern 3, 1923, pp. 231–239. [23] O.B. Pedersen, L.M. Brown, W.M. Stobbs, The Bauschinger effect in copper, Acta Metall. 29 (1981) 1843–1850. [24] H. Mughrabi, Dislocation wall and cell structures and long-range internal stresses in deformed metal crystals, Acta Metall. 31 (1983) 1367–1379. [25] J. Polák, M. Klesnil, The hysteresis loop 1. A statistical theory, Fatigue Eng. Mater. Struct. 5 (1982) 19–32. [26] J. Polák, M. Klesnil, J. Helešic, The hysteresis loop 2. An analysis of the loop shape, Fatigue Eng. Mater. Struct. 5 (1982) 33–44. [27] G. Miyamoto, J.C. Oh, K. Hono, T. Furuhara, T. Maki, Effect of partitioning of Mn and Si on the growth kinetics of cementite in tempered Fe-0.6 mass%C martensite, Acta Mater. 55 (2007) 5027–5038. [28] S. Ghosh, Rate-controlling parameters in the coarsening kinetics of cementite in Fe0.6C steels during tempering, Scr. Mater. 63 (2010) 273–276. [29] S. Allain, O. Bouaziz, M. Takahashi, Toward a new interpretation of the mechanical behaviour of as-quenched low alloyed martensitic steels, ISIJ Int. 52 (n.4) (2012) 717–722. [30] A. Arlazarov, O. Bouaziz, A. Hazotte, M. Gouné, S. Allain, Characterization and modeling of manganese effect on strength and strain hardening of martensitic carbon steels, ISIJ Int. 53 (n.6) (2013) 1076–1080. [31] W.C. Leslie, G.C. Rauch, Precipitation of carbides in low-carbon Fe-Al-C alloys, Metall. Trans. A 9A (1978) 343–349.
Table 7 Statistical performance of the models. YS0.2
Accuracy Bias Correlation Efficacy
UTS
1st model
2nd model
1st model
2nd model
78.2 159.9 0.85 0.86
65.8 43.4 0.94 1.23
101.1 −73.0 0.92 1.18
41.4 −37.0 0.99 1.07
tempering (σmin variable). 2) Secondly, if the sample is in the low tempering condition it is necessary to find the σmin that will be used. On the other hand, if the sample is in the high tempering condition, it is necessary to obtain the σminc and calculate σmin. The higher the value of low tempering limit and the σmin for low tempering, higher the value of σminc. The yield strength and the ultimate tensile strength predicted by both models are in agreement with the experimental values. However, improvements to model can be further developed to better predict these values. As equations, required in the second model, relating the constants to the composition are not available, future investigations are planned in order to include in the model the effect of alloying elements on the softening of tempered martensite. Acknowledgements The authors wish to express their gratitude to Virginie Blesse for her help in the experimental part of this work. References [1] R. Maddin, A history of martensite: some thoughts on the early hardening of iron, in: G.B. Olson, W.S. Owen (Eds.), Martensite, ASM Int., Materials Park, OH, 1992, pp. 11–19. [2] C.S. Smith, A history of martensite: early ideas on the structure of steel, in: G.B. Olson, W.S. Owen (Eds.), Martensite, ASM Int., Materials Park, OH, 1992, pp. 21–39. [3] L. Kaufman, M. Cohen, Thermodynamics and kinetics of martensitic transformations, Prog. Met. Phys. 7 (1958) 165–246. [4] C.M. Wayman, Introduction to the Crystallography of Martensite Transformations, MacMillian, New York, NY, 1964. [5] G. Krauss, A.R. Marder, The morphology of martensite in iron alloys, Metall. Trans. 2 (1971) 2343–2357. [6] A.R. Entwisle, The kinetics of martensite formation in steel, Metall. Trans. 2 (1971) 2395–2407.
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